On mathematical models subject to

homogeneous-heterogeneous reactions

By

ZAKIR HUSSAIN

Department of Mathematics

Quaid-I-Azam University

Islamabad, Pakistan

2018

On mathematical models subject to

homogeneous-heterogeneous reactions

By

ZAKIR HUSSAIN

Supervised By

PROF. DR. TASAWAR HAYAT

Department of Mathematics

Quaid-I-Azam University

Islamabad, Pakistan

2018

On mathematical models subject to

homogeneous-heterogeneous reactions

By

ZAKIR HUSSAIN

A THESIS SUBMITTED IN THE PARTIAL FULFILMENT OF THE

REQUIREMENT FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN

MATHEMATICS

Supervised By

PROF. DR. TASAWAR HAYAT

Department of Mathematics

Quaid-I-Azam University

Islamabad, Pakistan

2018

Dedicated To

My Parents

And

Supervisor

Author’s Declaration

I Zakir Hussain hereby state that my PhD thesis titled On mathematical

models subject to homogeneous-heterogeneous reactions is my own

work and has not been submitted previously by me for taking any degree from the

Quaid-I-Azam University Islamabad, Pakistan or anywhere else in the

country/world.

At any time if my statement is found to be incorrect even after my graduate the

university has the right to withdraw my PhD degree.

Name of Student: Zakir Hussain

Dated: 10-07-2018

Plagiarism Undertaking

I solemnly declare that research work presented in the thesis titled “On mathematical models

subject to homogenous-heterogeneous reactions” is solely my research work with no

significant contribution from any other person. Small contribution/help wherever taken has been

duly acknowledged and that complete thesis has been written by me.

I understand the zero tolerance policy of the HEC and Quaid-I-Azam University towards

plagiarism. Therefore, I as an Author of the above titled thesis declare that no portion of my thesis

has been plagiarized and any material used as reference is properly referred/cited.

I undertake that if I am found guilty of any formal plagiarism in the above titled thesis even

afterward of PhD degree, the University reserves the rights to withdraw/revoke my PhD degree

and that HEC and the University has the right to publish my name on the HEC/University Website

on which names of students are placed who submitted plagiarized thesis.

Student/Author Signature: a

Name: Zakir Hussain

Acknowledgments

All praise and thanks to Allah Almighty, the creator of this universe, who inculcated

in me the strength and spirit to fulfill the mandatory requirements for the completion of

this dissertation, so that today I can stand with my head held high. May Allah’s peace

and blessing be upon our Beloved Prophet Muhammad (PBUH) who was a mercy upon

us from Allah, whose character and nobility none has seen before or after Him (PBUH).

All my admiration goes to Him. May Allah give us all the ability to take heed from His

Seerah such that we desire to live our lives guided by His the Sunnah. My lengthy list of

acknowledgment comprises of all the important people whom Allah has blessed me with to

help with and contribute to this dissertation. I take this opportunity to acknowledge them

all and extend my sincere gratitude for helping me make this thesis a possibility.

First and foremost, I express immeasurable gratitude to all my teachers whose knowledge

and wisdom have brought me to this stage of academic zenith, but in particular, I owe an

immense debt of gratitude to the person who made the biggest difference in my life, my

honorable Supervisor, Chairman department of mathematics, Prof. Dr. Tasawar Hayat.

He has been a living role model to me, taking up new challenges every day, tackling them

with all his grit and determination and always thriving to come out victorious. I consider

myself fortunate enough to have such a good supervisor who can originally inspire hope,

ignite imaginations and instil love of learning. Without his patience, encouragement and

insightful suggestions, I could not finish my research work so smoothly and perfectly. To

me, professionally, he is ”perfection personified”.

I am grateful to my respected teachers Dr. Muhammad Ayub, Dr. Sohail Nadeem, Dr.

Masood Khan and Dr. Malik Muhammad Yousaf for their valuable suggestions in all

aspects.

I can not put aside the financial support of Prime Minister’s Fee Reimbursement Scheme

and Marafie Foundation at this moment of accomplishment. I also appreciate the sup-

port and cooperation of non-teaching staff department of mathematics, Mr. Zahoor, Mr.

Sheraz sahab, Mr. Bilal, Mr. Safdar and Mr. Sajid.

This acknowledgment will surely remain incomplete if I wouldn’t express my deep in-

debtedness and cordial thanks to the people who mean world to me, my family. I enact

my heartfelt thanks and respect, which springs from my soul to my father Abdullah and

mother Zarina, whose prayers are accompaniment in the journey of my life. Each and ev-

ery credit goes to them for making me what I am today. I am in great debt to acknowledge

the heartstrings and prayers of my caring sisters Sara, Zakia, Batool, Farzana, Samera

and my younger brother Meraj Hussain. I don’t imagine a life without their love and

blessings. Thank you all for having faith in me.

I extend my sincere word of thanks and my heartfelt gratitude to my Uncle Ghulam Has-

san Mehtab, who stood by me in all times whenever I needed him. I would never be able

to pay back the love and affection showered upon me by him from childhood up till now.

Worth mentioning here would be my best friend Muhammad Aslam, who selflessly and

whole-heartedly guided, helped and supported me throughout my PhD period.

My heartfelt thanks to Mr. Shakeel Ahmed, Mr. Habib Hassan, Ms. Nosheen Zahra,

Ms. Amna Mehdi, Dr. Muhammad Farooq, Dr. Taimoor Salahuddin, Dr. Masood Ur

Rahman, Dr. Ramzan Ali, Dr. Muhammad Waqas, Dr. Hashim, Dr. Taseer Muham-

mad, Dr. Muhammad Azam, Dr. Atta Ullah, Dr. Tehseen Abbas, Dr. Muhammad

Zubair, Dr. Shahid Farooq, Dr. Arifullah, Dr. Fahim Ud Din, Dr. Sardar Bilal, Mr.

Arif Hussain, Mr. Muhammad Naeem, Mr. Nadeem, Mr. Waleed khan, Mr. Ijaz

khan, Mr. Faisal Shah, Mr. Sajid Qayyum, Mr. Ikram Ullah, Mr. Latif Ahmad, Mr.

Jawad Ahmed, Mr. Bilal Ahmad, Mr. Khursheed Muhammad, Mr. Arsalan Aziz,

Mr. Zaheer Kiyani, Mr. Khalil ur Rahman, Mr. Amir, Mr. Sohail Ahmed, Mr. Us-

man Ali and Mr. Sajjad Hussain for always being there and bearing with me the good

and bad times during my wonderful days of PhD. I also thanks to all those people who

directly or indirectly helped me during my PhD journey. I also acknowledge my hostel

friends for keeping a healthy atmosphere and for being with me in thick and thins of life. I

consider myself lucky to be a member of QAU Athlete Team, Hockey team and Football

team. I also gratefully acknowledge the whole teams over here.

I especially deem to express my unbound thanks to all my friends especially Zahid Nisar,

Fakhar Abbas and Zahid Ahmed, who have lived by example to make me understand

the hard facts of life. I couldn’t have asked for more than what I got from them throughout

the period of M.Sc. M. Phil and PhD.!! Unforgettable memories...Thank you for the good

time we have all together.

The contributions vary but the appreciation is still large thus I leave it in the hands of Allah

to repay the debt to all those beautiful people. May Allah make all our intentions sincere

for His pleasure alone (Aameen).

July 10, 2018

Zakir Hussain

Nomenclaturekf thermal conductivity of fluid in viscous casek thermal conductivity of fluid in non-Newtonian caseknf thermal conductivity of nanoliquidl characteristics lengthhf heat transfer coefficientkCNT thermal conductivity of carbon nanotubese(u) eccentricity of ellipsoidg2 emperical parameterL1 length of CNTsR1 radius of CNTst1 thickness of interfacial layerθ1 angle between the axial directions of CNTsψ the sphericity of CNTsf1 Maxwell reflection coefficientb2 thermal accommodation coefficientd1 concentration accommodation coefficientkB Boltzmann constantp pressureξ1,ξ2 mean free path constantsΓ ratio of specific heatsa,b concentrations species A, B respectivelya0 positive constanta∗, b∗, c, d constantsA, B chemical speciesks, kr rate constantsDA, DB Diffusion coefficients of chemical speciesT temperature of fluidTw surface temperaturef stream functionT∞ ambient temperature of liquidTf temperature of hot fluidTm∗ melting temperatureλ3, Cs laten heat of fluid and heat capcity of fluid respectivelyR radius of cylinderγ curvature parameterk∗ permeability of porous mediumα∗1 second grade parameterβ0 strength of applied magnetic fieldσ1 electric conductivity

i

NomenclatureQ(z) nonuniform heat generation/absorptionQ0 the coefficient of heat generation/absorption per unit volumeU0 reference velocityUw nonlinear stretching velocityUe free stream velocitya1 dimensionless constantb1 dimensionless constantB small constantϕ∗ porosity of porous mediumg1 gravitational accelerationcb drag coefficientβ2, c1 fluid parameters of Powell-Eyring modelk1 permeability parameterh∗1 dimensional velocity slip parameterh∗2 dimensional temperature jump parameterλ∗1, λ

∗2, relaxation and retardation times respectively

βT thermal expansion coefficientCf (T∞−Tm∗ )

L∗m

Stefan number for liquidCs(Tm∗−T0)

L∗m

Stefan number for solidL∗m laten heat of melting

m shape parameterα wall thickness parameterDa inverse Darcy numberβ local inertia parameterA ratio of free stream velocity to stretching velocityh1 velocity slip variableα1 ratio parameterω viscoelastic parameterHa Hartman numberλ1,M1 fluid parametersh2 dimensionless temperature jump parameterδ heat generation parameterα2 conjugate parameter for Newtonian heatingM melting parameterS thermal stratification parameterϵ Weissenberg numberEc Eckert numberSc Schmidt numberK homogeneous reaction parameterKs heterogeneous reaction parameterBi Biot numberδ1 ratio of diffusion coefficients

ii

NomenclatureRez local Reynolds number(u, w) velocity components in r − z plane(u, v) velocity components in x− y plane(u,v,w) velocity components in x− y − z directions(r, z) cylindrical coordinates(x, y) cartesian coordinateshf ,hθ,hζ auxiliary parameterswe velocity of stretching cylinderNuz local Nusselt numberCf skin friction coefficientτw surface shear stressqw surface heat fluxβ3, c1 Powell-Eyring fluid parametersβ1, β2 Deborah numbers via relaxation and retardation timesλ mixed convection variableGrx Grashof number

Greek symbolsµf dynamic viscosity of liquidρf density of fluidνf kinematic viscosity of liquidcp specific heat via constant pressure(cp)nf specific heat via constant pressure of nanofluidµnf dynamic viscosity of nanofluidνnf kinematic viscosity of CNTs liquidαnf thermal diffusivity of nanofluidα∗ thermal diffusivity of fluid(cp)CNT specific heat via constant pressure of CNTs(ρ)CNT density of CNTsρnf density of nanofluidϕ volume fraction of nanomaterialsθ temperature in dimensionless formζ dimensionless concentrationη transformation parameter

iii

AbbreviationsOHAM optimal homotopy analysis methodHAM homotopy analysis methodBvp boundary value problemSWCNTs single-walled carbon nanotubesMWCNTs multi-walled carbon nanotubesCNTs carbon nanotubesWater base fluidKerosene oil base fluidSWCNT-Water nanofluid with single wall carbon nanotubesMWCNT-Water nanofluid with multi wall carbon nanotubeSWCNT-Kerosene oil nanofluid with single wall carbon nanotubeMWCNT-Kerosene oil nanofluid with multi wall carbon nanotubeH-C model Hamilton-Crosser modelLU Lower UpperPR present result

iv

Preface

Nanoliquids strengthen low thermal conductivity of materials. Nanoliquid consists of

nano-material (1− 100 nm) and base-liquid. Nanoliquids are regarded functional in engi-

neering, electronic process and many other fields. Nanoparticles include CNTs (MWCNT,

SWCNT), oxides and carbides ceramics and semiconductors. These nanoparticles are sub-

merged in an ordinary fluid to make them nanofluids. Non-Newtonian fluids like Oldroyd-

B, Powell Eyring, Williamson are regarded helpful in physiological phenomena, pharma-

ceutical process, paper production and metallurgy. That is why Oldroyd-B, Powell Eyring

and Williamson fluids are adopted in this thesis for modeling and analysis of flows in

boundary layer region. The boundary-layer flows due to stretching surface have wide

range of applications in industries and engineering. Further it is also taken into account

that heterogeneous-homogeneous reactions in liquid flow have vital role following com-

bustion, biochemical processes, catalysis and in many other fields. Keeping all these as-

pects in mind the prime objective of this thesis is to study nonlinear mathematical models

subject to homogeneous-heterogeneous reactions. The structure of this thesis is as follows.

Chapter 1 contains literature survey and some basic conservation laws. Mathematical

model and boundary-layer expressions for Oldroyd-B, second grade, Powell Eyring and

Williamson fluids are incorporated. Five different techniques are used to deal with the

flow problems. Thus basic concepts homotopy analysis method (HAM), Bvp4c matlab

solver, Optimal homotopy analysis method (OHAM), shooting technique and Keller box

method are provided.

Chapter 2 addresses the impact of diffusion species in flow of CNTs nanofluid sat-

urating porous medium. Melting heat transfer is present. Auto catalyst and reactant

have same diffusion coefficients. Flow induced by stretched cylinder. Homotopy anal-

v

ysis method (HAM) is adopted for solutions procedure. The outcomes for CNTs flow are

disclosed. This chapter contents is reported in Journal of Molecular Liquids 221 (2016)

1121− 1127.

Chapter 3 deals diffusion species via CNTs with convective conditions. Flow generated

is because of stretching cylinder. OHAM is adopted for outcomes. Graphical outcomes

are discussed via variables for flow. The outcomes are reported in Journal of the Taiwan

Institute of Chemical Engineers 70 (2017) 119− 126.

Chapter 4 reports computational aspects for Forhheimer flow of CNTs nanofluids with

diffusion species. In this chapter thermal conductivity of CNTs nanofluid is compared via

renovated Hamilton-Crosser (H-C) and Xue models for flows by stretching cylinder and

flat sheet. The results are obtained via Keller box method. The findings of this chapter are

submitted in Physica E for possible publication.

Chapter 5 examines stagnation flow of carbon water and carbon kerosene oil nanofluids

via nonlinear stretched surface. CNTs nanofluids fill the porous medium. Homogeneous-

heterogeneous reactions and melting effects are considered. Outcomes are obtained via

(OHAM). Heat transferred is addressed via different variables involved in solutions ex-

pressions. The contents of this chapter are published in Advanced Powder Technology

27 (2017) 1677− 1688.

Chapter 6 presents 3D nanoliquid flow by stretched (nonlinear) sheet with diffusion

species. Nanoliquid is saturated via porous space. Convective condition and heat source/sink

are used for heat mechanism. The numerical outcomes are analyzed via shooting approach.

Graphical illustrations and tabulated values are disclosed. The main findings of this chap-

ter can be seen through Computer Methods in Applied Mechanics and Engineering 329

(2018) 40− 54

vi

Chapter 7 describes three-dimensional (3D) nanoliquid flow via slendering stretch-

ing (nonlinear) sheet with slip effects. bvp4c technique is used for numerical outcomes.

Tabulated and graphical findings are explored via sundry variables. These contents are

published in Computer Methods in Applied Mechanics and Engineering 319 (2017)

366− 378.

Chapter 8 discloses flow of MHD non-Newtonian liquid via Newtonian heating and

diffusion species. Results are developed via HAM. Numerical results of skin friction and

Nusselt number are disclosed. The findings of current chapter are published in PloS one

11 (6) e0156955 (2016).

Chapter 9 describes flow of MHD viscoelastic fluid with species. Flow is due to

stretched cylinder. Viscous dissipation, Newtonian heating and Joule heating are also ac-

counted. The results are constructed via HAM. Characteristics of different variables are

elaborated graphically. The outcomes of this chapter are published in Journal of Mechan-

ics 33 (2017) 77− 86.

Chapter 10 includes species influence in flow of second grade material. Melting heat

contribution is inspected. Inclined magnetic line is used to electrified the liquid. Numerical

results are addressed via heat transfer and skin friction. The contents are addressed in

Journal of Molecular Liquids 215 (2016) 749− 755.

Chapter 11 investigates homogeneous-heterogeneous reactions in thermally stratified

stagnation flow of viscoelastic liquid with mixed convection. Flow is by a stretched sheet.

Influences of various variables on quantities of interest are discussed. The findings of this

chapter are published in Results in Physics 6 (2017) 1161− 1167.

Chapter 12 addresses convective flow of Williamson fluid by cylinder and flat sheet.

Convective condition is used for heat transfer mechanism. The species of auto-catalyst

vii

and reactant are used to regulate the concentration. Convection or evaporation for tem-

perature phase change is analyzed through homogeneous-heterogeneous reactions. The

transformed ordinary differential equations are dealt numerically via Keller box method.

Impacts of pertinent parameters of interest are graphically discussed. Comparison of re-

sults for cylinder and flat sheet is arranged. The contents of this chapter are submitted for

possible publication in International Journal of Mechanical Sciences.

viii

Contents

List of Tables xvi

List of figures xxiv

1 Background and basic laws 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Fundamental laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Conservation law of mass . . . . . . . . . . . . . . . . . . . . . 4

1.3.2 Conservation law of linear momentum . . . . . . . . . . . . . . . 4

1.3.3 Energy conversation . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.4 Conservation law of concentration . . . . . . . . . . . . . . . . . 5

1.4 Viscous liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Non-Newtonian liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5.1 Second grade liquid . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5.2 Powell Eyring liquid . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5.3 Oldroyd-B liquid . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5.4 Williamson liquid . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Solution methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ix

1.6.1 Homotopy analysis method . . . . . . . . . . . . . . . . . . . . . 9

1.6.2 Optimal homotopy analysis method . . . . . . . . . . . . . . . . 9

1.6.3 Bvp4c Matlab solver . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6.4 Shooting technique . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6.5 Keller box method . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Flow of Carbon nanotubes with melting heat transfer 11

2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Homotopic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Main outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Diffusion species in convective CNTs flow through a permeable

space 28

3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 OHAM outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Computational study for CNTs nanofluid with renovated Hamilton-

Crosser and Xue models past a stretching cylinder 43

4.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Keller-box results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.1 Reduction of the nth order system to nth 1st order equations . . . 48

4.2.2 Finite difference discretization . . . . . . . . . . . . . . . . . . . 48

4.2.3 Quasilinearization of non-linear Keller algebraic equations . . . . 50

x

4.2.4 The Block tridiagonal matrix . . . . . . . . . . . . . . . . . . . . 51

4.2.5 Block-tridiagonal elimination of linear Keller algebraic equations 53

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Stagnation point in CNTs flow 65

5.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 OHAM outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Convective flow of Carbon nanotubes via three-dimensional 82

6.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.2 Shooting technique results . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.4 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 CNTs flow with slip condition 102

7.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2 Bvp4c outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.4 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8 Newtonian heating flow with diffusion species 116

8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.2 Homotopic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xi

8.4 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9 Joule heating and viscous dissipation in chemical reactive flow 134

9.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9.2 Homotopy results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.2.1 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . 137

9.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

10 Diffusion species in viscoelastic liquid flow with melting heat 147

10.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

10.2 HAM outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

10.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

10.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

11 Influence of diffusion species in thermal an Oldroyd-B liquid flow

163

11.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

11.2 HAM outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

11.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

11.4 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

12 Numerical simulation for chemical species and Joule heating in

MHD flow of Williamson fluid 175

12.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

12.2 Implicit finite difference scheme . . . . . . . . . . . . . . . . . . . . . . 179

12.3 The block tridiagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . 185

xii

12.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

12.5 Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

xiii

List of Tables

2.1 Outcomes for CNTs liquid [99]. . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Convergence of equations via γ = 0.2, k1 = 0.1, M = 0.1, Ks = 1.2,

K = 0.4, ϕ = 0.1 and Sc = 1.5. . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Numerical results of individual residual errors via SWCNTs liquid and

MWCNTs liquid at different order with γ = 0.1, k1 = 0.1, ϕ = 0.1,

Bi = 0.1, K = 0.4, Ks = 1.2 and Sc = 1.5. . . . . . . . . . . . . . . . . 41

3.2 Convergence via γ = 0.2, k1 = 0.1, Ks = 0.1, K = 0.2, Bi = 0.2,

ϕ = 0.2 and Sc = 1.3 for the series solutions. . . . . . . . . . . . . . . . 41

3.3 Comparison of f ′′(0) via k1 when γ = 0, ϕ = 0 [97]. . . . . . . . . . . . 42

4.1 Skin friction for various values of ϕ, β, Da. . . . . . . . . . . . . . . . . 62

4.2 Nusselt number for ϕ in case of SWCNTs liquid and MWCNTs liquid. . . 63

4.3 Skin friction and Nusselt number for CNTs liquid via various values of

curvature parameter γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Validation of skin friction [99] via Da = 0 and β = 0 for ϕ. . . . . . . . 63

5.1 Average square residual errors via M = α = A = ϕ = k1 = 0.1, m = 2,

K = 0.4, Ks = 0.9 and Sc = 1.2. . . . . . . . . . . . . . . . . . . . . . 72

xiv

5.2 Validation of f ′′ (0) with [100], [101] and [102] via A when k1 = ϕ =

Ks = K = Sc = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Validation of f ′′(0) via k1 when A = 0, ϕ = 0 [97]. . . . . . . . . . . . . 72

6.1 Numerical values of skin frictionRe12xCfx andRe

12yCfy for SWCNTs liquid

and MWCNTs liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2 Resutls for Re−12Nux via ϕ, δ and γ in case of CNTs liquid. . . . . . . . 100

6.3 Validation of f ′′(0) via k1 and ϕ = 0, [97], [104]. . . . . . . . . . . . . . 100

7.1 Skin friction and Nusselt number for CNTs liquid via ϕ, k1, α, h1 and h2. 114

7.2 Validation [97], [104] of f ′′(0) for k1 fixed h1 = 0, h2 = 0, ϕ = 0, α = 0,

n = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.1 Convergence via solutions at different order by fixing γ = M1 = λ1 =

Ha = α2 = δ = 0.1, K = 0.6, Ks = 1.0, Sc = 1.2, and Pr = 0.7. . . . . 131

8.2 Validation of f ′′(0) [107] via γ = Ha = 0. The present results are closed

in brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.3 Validation of skin friction coefficient Re1/2z Cf [107] when γ =Ha = 0.

The present results are in brackets . . . . . . . . . . . . . . . . . . . . . 132

8.4 Nusselt number via different variables. . . . . . . . . . . . . . . . . . . . 132

8.5 CfRe1/2z via variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.6 Validation of f ′′(0) when γ =λ1= M1 = 0. . . . . . . . . . . . . . . . . . 133

9.1 Series solutions convergence when γ = ω = 0.1, Ks = 1.2, Ec = 0.1,

Sc = 0.9, α2 = Ha = 0.1, K = 0.4 and Pr = 0.8. . . . . . . . . . . . . 145

9.2 Comparison of f ′′(0) by varying Ha and putting γ = 0, ω = 0 [110]. . . . 145

9.3 Skin friction via variable. . . . . . . . . . . . . . . . . . . . . . . . . . . 145

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9.4 Nusselt number via different parameters. . . . . . . . . . . . . . . . . . . 146

10.1 Convergence for outcomes via γ = M = ω = Ha = δ = 0.1, K = 0.4,

Ks = 1.2, Φ = π4, Sc = 0.9and Pr = 1.2. . . . . . . . . . . . . . . . . . 151

10.2 Skin friction via parameters. . . . . . . . . . . . . . . . . . . . . . . . . 161

10.3 Outcomes of Nusselt number via variables. . . . . . . . . . . . . . . . . 161

11.1 Convergence for the solutions via β1 = β2 = A = λ = S = 0.1, Pr = 1,

K = 0.4. Ks = 0.9, Sc = 1.2 and ~ = −1.2. . . . . . . . . . . . . . . . 166

11.2 Comparison of f ′′(0) via β1, β2 and A when λ = 0 through the refs.

[38, 114, 115, 113]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

11.3 HAM outcomes and Bvp4c outcomes via λ = 0.1, β2 = 0.1, A = 0.1,

S = 0.1, Pr = 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

12.1 Comparison of −f ′′(0) in limiting case via γ when ϵ = Ha = 0. . . . . . 207

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List of Figures

2.1 Geometry of problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 ~-curve for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 ~-curves for f(η) and ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Plots via ϕ for f ′(η) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Plots via k1 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Plots via M for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Plots via γ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.8 Plots via M for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Plots via γ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.10 Plots via γ for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.11 Plots via M for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.12 Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.13 Plots via Ks for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.14 Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.15 Plots via ϕ and k1 for skin friction coefficient. . . . . . . . . . . . . . . 24

2.16 Plots via γ and k1 for skin friction coefficient. . . . . . . . . . . . . . . 25

2.17 Plots via ϕ and k1 for Nussetl number. . . . . . . . . . . . . . . . . . . . 25

2.18 Plots via M and γ for Nussetl number. . . . . . . . . . . . . . . . . . . 26

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3.1 Geometry of problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Plots via k1 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Plots via γ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Plots via ϕ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Plots via Bi for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6 Plots via γ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Plots via γ for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.9 Plots via Ks for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.10 Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.11 Plots for skin friction via γ and k1. . . . . . . . . . . . . . . . . . . . . 39

3.12 Plots for Nusselt number ϕ via γ. . . . . . . . . . . . . . . . . . . . . . 40

4.1 Physical model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Net ”Keller box” for different approximations. . . . . . . . . . . . . . 49

4.3 Plots via β for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Plots via Da for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5 Plots via ϕ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6 Plots via γ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.7 Plots via ϕ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.8 Plots via γ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.9 Plots via ϕ for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.10 Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.11 Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.12 Plots for skin friction via γ and ϕ. . . . . . . . . . . . . . . . . . . . . 61

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4.13 Plots for Nusselt number via γ and ϕ. . . . . . . . . . . . . . . . . . . 62

5.1 Geometry of problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Total errors via SWCNTs liquid (a) and kerosene SWCNTs liquid (b). 71

5.3 Total errors via MWCNTs water liquid (a) and kerosene MWCNTs

liquid (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4 Plots via α with m = 0.5 for f ′ (η). . . . . . . . . . . . . . . . . . . . . 75

5.5 Plots via α with m = 5 for f ′ (η). . . . . . . . . . . . . . . . . . . . . . 75

5.6 Plots via k1 for f ′ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.7 Plots via ϕ for f ′ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.8 Plots via A for f ′ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.9 Plots via M for θ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.10 Plots via ϕ for θ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.11 Plots via A for θ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.12 Plots via K for ζ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.13 Plots via Ks for ζ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.14 Plots via Sc for ζ (η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.15 (a) Plots for skin friction, (b) Plots for Nusselt number. . . . . . . . . 80

6.1 Physical coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 Plots via ϕ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3 Plots via k1 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.4 Plots via n for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.5 Plots via α1 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.6 Plots via δ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.7 Plots via Bi for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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6.8 Plots via ϕ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.9 Plots via α1 for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.10 Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.11 Plots via Ks for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.12 Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.13 Outlines for Xue and renovated H-C models. . . . . . . . . . . . . . . 96

6.14 Plots via Bi and ϕ for Xue and renovated H-C models. . . . . . . . . . 97

6.15 Plots of streamlines for SWCNTs. . . . . . . . . . . . . . . . . . . . . 97

6.16 Plots of streamlines for MWCNTs. . . . . . . . . . . . . . . . . . . . . 98

6.17 Plots of isotherms for SWCNTs. . . . . . . . . . . . . . . . . . . . . . 98

6.18 Plots of isotherms for MWCNTs. . . . . . . . . . . . . . . . . . . . . . 99

7.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.2 Plots via ϕ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3 Plots via ϕ for g′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4 Plots via k1 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.5 Plots via k1 for g′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.6 Plots via h1 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.7 Plots via h1 for g′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.8 Plots via α for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.9 Plots via α for g′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.10 Plots via h1 for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.11 Plots via h2 for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.12 Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.13 Plots via Ks for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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7.14 Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.1 Geometry of problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.2 ~-curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.3 Plots via Ha for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.4 Plots via γ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.5 Plots via M1 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.6 Plots via Ha for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.7 Plots via γ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.8 Plots via M1 for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.9 Plots via Pr for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.10 Plots via α2 for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.11 Plots via δ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.12 Plots via Ha for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.13 Plots via γ for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.14 Plots via M1 for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.15 Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.16 Plots via Ks for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.17 Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9.1 ~-curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.2 Plots via γ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9.3 Plots via ω for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9.4 Plots via γ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

9.5 Plots via ω for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

9.6 Plots via α2 for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

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9.7 Plots via Pr for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.8 Plots via γ for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.9 Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.10 Plots via Ksfor ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.11 Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

10.1 ~−curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

10.2 ~−curves for residual error ∆fk . . . . . . . . . . . . . . . . . . . . . . 153

10.3 ~−curves for residual error ∆θk. . . . . . . . . . . . . . . . . . . . . . 153

10.4 ~−curves for residual error ∆ζk. . . . . . . . . . . . . . . . . . . . . . 154

10.5 Plots via γ for f(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

10.6 Plots via ω for f(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.7 Plots via Ha for f(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.8 Plots via M for f(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

10.9 Plots via Φ for f(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

10.10Plots via γ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10.11Plots via δ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10.12Plots via Pr for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

10.13Plots via M for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

10.14Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10.15Plots via Ks for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10.16Plots for Nusselt number via γ and M . . . . . . . . . . . . . . . . . . . 160

10.17Plots for Nusselt number via γ and Pr. . . . . . . . . . . . . . . . . . 160

11.1 ~−curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

11.2 Plots via A for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

xxii

11.3 Plots via λ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

11.4 Plots via β1 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

11.5 Plots via β2 for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

11.6 Plots via S for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

11.7 Plots via Pr for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

11.8 Plots via β2 for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

11.9 Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

11.10Plots via Ks for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

11.11Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

11.12Plots for Nusselt number via S and Pr. . . . . . . . . . . . . . . . . . 173

11.13Plots for Nusselt number via λ and S. . . . . . . . . . . . . . . . . . . 173

12.1 Schematic representation of problem. . . . . . . . . . . . . . . . . . . 176

12.2 Net Keller box for finite difference approximation. . . . . . . . . . . . 181

12.3 Plots via ϵ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

12.4 Plots via Ha for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

12.5 Plots via γ for f ′(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

12.6 Plots via Ha for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

12.7 Plots via Pr for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

12.8 Plots via Bi for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

12.9 Plots via δ for θ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

12.10Plots via K for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

12.11Plots via Ks for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

12.12Plots via Sc for ζ(η). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

12.13Plots for skin friction coefficient via ϵ and Ha. . . . . . . . . . . . . . 205

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12.14Plots for skin friction coefficient via Ha and γ. . . . . . . . . . . . . . 205

12.15Plots for Nusselt number via Pr and γ. . . . . . . . . . . . . . . . . . 206

12.16Plots for Nusselt number via Pr and Bi. . . . . . . . . . . . . . . . . . 206

12.17Plots for Nusselt number via Ec and δ. . . . . . . . . . . . . . . . . . . 207

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Chapter 1

Background and basic laws

1.1 Introduction

Review of some studies associated to boundary layer flow, non-Darcy Forchheimer,

mixed convection, stagnation point flow, heat and mass transfer, nanofluids, Newtonian

heating, melting heat transfer and diffusion species reactions are incorporated here. Math-

ematical modeling for viscous, Oldroyd-B fluid, viscoelastic fluid, Powell-Eyring fluid and

Williamson fluid are addressed for better understanding of upcoming chapters. The solu-

tion methodologies like Homotopy analysis method (HAM), Optimal homotopy analysis

method (OHAM), bvp4c Matlab solver, Shooting technique and Implicit finite difference

method (Keller box method) are briefly explained in present chapter.

1.2 Background

MWCNTs (i.e. Multi walled carbon nanotubes) introduced first time via Krashtschmer

and Huffman technique by Lijima in 1991. SWCNTs (Single walled carbon nanotubes)

has been reported in 1993 by Donald Bethune. CNTs (i.e. Carbon nanotubes) are exten-

sively used in health care, environment, electronics and in many other areas. The issues

1

associated with thermal conductivity (i.e water, oil, ethylene glycol, gasolene etc) can be

enhanced by nanolquids [1-8]. It is investigated that nanoliquids have higher thermal prop-

erties when compared with ordinary liquids. Copper nanoparticles are added in liquid by

Choi [9] and noticed higher thermal conductivity. More reviewed on nanoliquids are dis-

cussed in these works [10-19]. Natural convective flow of nanoliquid in 3D is reported by

Sheikholeslami et al. [20]. MHD radiative flow with convective boundary condition and

prescribe surface heat flux is highlighted by Mahanthesh et al. [21-22]. Convective flow of

nanoliquid over deformable stretching sheet is reported by Hayat et al. [23] and Khan et

al. [24]. Third grade liquid flow with porous mechanism is investigated [25] ( Aziz et al.).

Radiative and convective flow of MHD nanoliquid is addressed by Sulochana et al. [26].

Ahmed et al. [27] reviewed convective MHD nanoliquids flow. Thermal stratification of

viscous liquid with convection is addressed by Mahmood et al. [28]. Babu et al. [29] re-

ported effects of cross-diffusion in MHD fluid flow. Convective flow of third grade liquid

with chemically reaction is investigated [30] (Hayat et al.). Babu et al. [31] addressed re-

active slip flow with variable heat source/sink. Reddy et al. [32] reported MHD ferrofluid

flow with frictional heating and radiation past a slendering stretched sheet.

Darcy law has wide utilization in geothermal processes, thermal engineering, petroleum

technology, chemical industries and in many other area. Darcy law is not applicable over

those region where the permeable mechanism has larger flow ratio due to non-uniform

near the wall region. Non-Darcian via permeable mechanism in flow and heat transfer rate

become necessary to examine. Tamayol et al. [33] reported heat investigation of liquid

flow by a permeable mechanism. Hong et al. [34] marked natural convection flow with the

impact of non-Darcy and nonuniform porosity. Khani et al. [35] examined heat transfer in

flow of third grade fluid filling non-Darcy porous media. Non-Darcian fluid flow under the

2

effect of thermal radiation is highlighted by Pal et al. [36]. Hayat et al. [37] studied CNTs

nanofluid flow in non-Darcy porous medium.

Analysis for non-Newtonian liquids flow is a subject of present thesis [38 − 52]. The

Williamson model [53] is also one amongst the non-Newtonian models. It explains the

flow of pseudoplastic materials. Obviously equations governing the flow of Williamson

fluid are higher order and more challenging than the Navier-Stokes expressions. Therefore

not much has been said about this fluid model. Few researchers made some attempts for

flows of Williamson fluid. Cramer et al. [54] experimentally studied the polymer melts

and particle suspensions by using Williamson fluid. Lyubimov et al. [55] analyzed gravity

effect in flow of Williamson fluid due to an inclined sheet. MHD Williamson liquid flow

via heat transfer is addressed by Hayat et al. [56]. One of the non-Newtonian fluids

is the Powell-Eyring liquid [57-61]. Second grade liquid [62-68] is the differential type

liquid. Normal stress can be discussed by this model. Industries and technologies have

wide range utilization of melting heat [69-73]. Heat and mass transport via convection are

major attention of researchers [74-78].

In current time human society highly needs the energy resources. Scientists have stim-

ulated in this area to develop advance energy resources and technologies [79-82]. Homo-

geneous and heterogeneous types are the two forms of reactions. Homogeneous reaction

tends to catalyst in one phase (same phase) while heterogeneous reaction occurs in two or

more different phases. Biochemical phenomenon, catalysis, combustion etc have been ac-

counted wide range of application of these reactions. Merkin et al. [83] analyzed chemical

species in flow via isothermal model. Few studies predicting homogeneous-heterogeneous

reactions in flows are mentioned by the refs. [84-95].

3

1.3 Fundamental laws

1.3.1 Conservation law of mass

Equation of continuity shows that mass neither be formed nor demolished. Mathemat-

ically it can be defined as:

∂ρf∂t

+∇ · (ρfV) = 0, (1.1)

where ρf signifies the density of liquid and V (= (u, v, w)) stands for velocity field.

In case of incompressible liquid Eq. (1.1) reduces to

∇ ·V = 0. (1.2)

In Cartesian coordinates system

∂u

∂x+∂v

∂y+∂w

∂z= 0, (1.3)

while in cylindrical coordinates becomes

1

r

∂

∂r(rur) +

1

r

∂

∂θ(vθ) +

∂

∂z(wz) = 0. (1.4)

1.3.2 Conservation law of linear momentum

Momentum remains conserved of whole system. Newton’s second law is used to derive

it. It is stated Mathematically as

ρfa = −∇p+ divτ + ρf f , (1.5)

4

where ρf denotes the fluid density, a the acceleration, p the pressure, τ the extra stress

tensor and f the body force per unit mass.

1.3.3 Energy conversation

It is developed via thermodynamics first law. It shows the total energy of system re-

mains conserved. Mathematically it can be represented as follows:

(ρc)fdT

dt= τ · L− divq− divqr. (1.6)

In above on L.H.S ((ρc)fdTdt) signifies internal energy, on R.H.S ((τ .L)) represents viscous

dissipation whereas ((divq)) and ((divqr)) on R.H.S indicate thermal and radiative heat

fluxes respectively. ρf , cf , τ stand for fluid density, specific heat at constant pressure

and liquid temperature. Further τ , (q,qr) symbolize Cauchy stress tensor and thermal,

radiative heat fluxes respectively. Fourier’s law of heat conduction and Stefan Boltzman

law are utilized to describe the thermal (q) and radiative heat (qr) fluxes.

1.3.4 Conservation law of concentration

It is addressed as;

dC

dt= −∇ · j. (1.7)

Using Fick’s law;

j = −D∇C. (1.8)

Hence the mass transport equation takes the form

dC

dt= D∇2C, (1.9)

5

in which C, D, j denote concentration of specie, mass diffusivity and characterizes mass

flux respectively.

1.4 Viscous liquid

Cauchy stress tensor for incompressible viscous fluid is as follows:

τ = −pI+ µA1 (1.10)

1.5 Non-Newtonian liquids

1.5.1 Second grade liquid

The equation for steady laminar second grade liquid flow is stated as:

∇ ·V = 0, (1.11)

ρdV

dt= divτ , (1.12)

where d/dt denotes the material derivative. The volume force is not accounted here.

The constitutive equation for the second grade fluid is

τ = −pI+ µA1 + α⋆1A2 + α⋆

2A21, (1.13)

A2 =dA1

dt+A1L+ LtA1, A1 = L+ Lt, L = ∇V, (1.14)

For consistency of thermodynamic analysis the conditions must hold i.e. µ ≥ 0, α⋆1 ≥ 0,

α⋆1 + α⋆

2 = 0.

6

L = (grad V)=

∂ur

∂r1r∂ur

∂θ− vθ

r∂vr∂z

∂vθ∂r

1r∂vθ∂θ

+ ur

r∂vθ∂z

∂wz

∂r1r∂wz

∂θ∂wz

∂z

,

∇ · τ =

∂τrr∂r

+ τrrr+ 1

rτrθ∂θ

+ ∂τzr∂z

− τθθr

∂τθr∂r

+ τθrr+ 1

rτθθ∂θ

+ τrθr+ ∂τθr

∂z

∂τzr∂r

+ τzrr+ 1

r∂τθz∂θ

+ ∂τzz∂z

1.5.2 Powell Eyring liquid

The stress tensor is

τ =

{µ+

1

βξ·sinh−1(

1

cξ·)

}A1, (1.15)

ξ· =

√1

2tr(A1)

2, (1.16)

sinh−1(1

cξ·) ≈ ξ·

c− ξ·3

6c3. (1.17)

Here in view of boundary layer approximations one has

∂u

∂r+u

r+∂w

∂z= 0, (1.18)

ρ

(u∂u

∂r+ w

∂u

∂z

)=∂τrr∂r

+∂τrz∂z

+τrrr, (1.19)

ρ

(u∂w

∂r+ w

∂w

∂z

)=

1

r

∂

∂r(rτrz) +

∂τzz∂z

, (1.20)

where u, w are the velocities in the r − z plane.

7

1.5.3 Oldroyd-B liquid

Here one considers

τ ij = −pδij + Sij (1.21)

where τ ij represents the components of Cauchy stress tensor, p the pressure, δij the com-

ponents of identity tensor and Sij the components of an extra stress tensor defined by

(1 + λ∗1

D

Dt

)Sij = µ

(1 + λ∗2

D

Dt

)Aij

1 , (1.22)

Aij1 the components of first Rivlin-Ericksen tensor and D

Dtthe contravariant convective

derivative.

1.5.4 Williamson liquid

The constitutive equations for Williamson fluid model are defined as follows;

τ =

(µ∞ +

µ0 − µ∞

1− Γγ

)A1, (1.23)

γ =

√Π

2, (1.24)

Π = trace (A1)2 , (1.25)

γ =

√(∂u

∂r

)2

+

(∂u

∂z+∂w

∂r

)2

+(ur

)2. (1.26)

Here µ∞ infinite shear rate viscosity, µ0 zero shear rate viscosity, γ deformation rate, Γ

time-dependent material constant and A1 first Rivlin Ericksen tensor. Considering µ∞ = 0

and Γγ < 1, one obtains

τ =

(µ0

1− Γγ

)A1. (1.27)

8

By utilizing binomial expansion

τ = µ0 (1 + Γγ)A1. (1.28)

τrr = 2µ0 [1 + Γγ]

(∂u

∂r

), τrz = µ0 [1 + Γγ]

(∂u

∂r+∂w

∂r

), (1.29)

τθθ = 2µ0 [1 + Γγ](ur

), τzr = µ0 [1 + Γγ]

(∂w

∂r+∂u

∂z

), (1.30)

1.6 Solution methodologies

1.6.1 Homotopy analysis method

This method deals highly nonlinear problems. The detail procedure of this method is

applied in chapter 8.

1.6.2 Optimal homotopy analysis method

The concept of minimization is used for average square residual errors.

εfk (hf ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη

]2, (1.31)

εθk (hf , hθ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη ,

k∑i=0

(θi)η=jπη

]2, (1.32)

εζk (hf , hζ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη ,

k∑i=0

(ζi)η=jπη

]2, (1.33)

εt = εfk + εθk + εζk. (1.34)

Here εt is total averaged squared residual error.

9

1.6.3 Bvp4c Matlab solver

Boundary value problem is

t′ = f(x, t,D), a ≤ x ≤ b (1.35)

with

h(t(a), t(b),D) = 0, (1.36)

D stands for unknown variables vector.

1.6.4 Shooting technique

Shooting technique deals only initial values problems. Thus modeled expressions are

converted into first order ODE’s.

1.6.5 Keller box method

Implicit finite difference scheme is utilized to solve the model equations. The details

are incorporated in Chapter 4.

10

Chapter 2

Flow of Carbon nanotubes with melting heat transfer

This chapter incorporates the diffusion species in flow of nanoliquid saturating porous

mechanism. Melting heat transfer is accounted in flow via stretched cylinder. Water is

utilized as the base liquid. Same diffusion coefficient is used via both auto catalyst and

reactant. Resulting differential systems are evaluated via HAM. Outcomes are examined

via graphs. The CfRe1/2z and NuzRe

−1/2z have been explored.

2.1 Formulation

CNTs Flow via stretched cylinder is contemplated. Melting heat and diffusion species

equations are dealt. Use Tm∗ < T∞. Here CNTs(SWCNTs and MWCNTs) are accounted

as nanoparticles in water base liquid. The heat produced via irreversible chemical reaction

is not accounted. Homogenous reaction via cubic auto catalysis can be addressed by

A+ 2B → 3B, rate = krab2. (2.1)

Isothermal equation of first succession is

11

Fig. 2.1: Geometry of problem

A → B, rate = ksa, (2.2)

The geometry of flow is developed via Fig. 2.1. The equations governing the flow are

∂ (ru)

∂r+∂ (rw)

∂z= 0, (2.3)

w∂w

∂z+u

∂w

∂r= νnf

(∂2w

∂r2+

1

r

∂w

∂r

)− νnf

k∗w, (2.4)

w∂T

∂z+u

∂T

∂r= αnf

(1

r

∂T

∂r+∂2T

∂r2

), (2.5)

u∂a

∂r+w

∂a

∂z= DA

(∂2a

∂r2+

1

r

∂a

∂r

)− krab

2,

u∂b

∂r+w

∂b

∂z= DB

(∂2b

∂r2+

1

r

∂b

∂r

)+ krab

2, (2.6)

12

the boundary conditions are stated as:

w = we =U0z

l, u = 0, T = Tm∗ ,

DB∂b

∂r= −ksa, DA

∂a

∂r= ksa, at r = R

a→ a0, b→ 0, w → 0, T → T∞, as r → ∞, (2.7)

knf

(∂T

∂r

)= ρnf (λ+ Cs (Tm∗ − T0))u, at r = R. (2.8)

Here Eq. (2.8) addresses heat transferred to melting surface is same with the melting heat

needed solid temperature (i.e. T ) to melting temperature (i.e. Tm∗).

Following Xue [96]

µnf =µf

(1− ϕ)5/2, ρnf = (1− ϕ) ρf + ϕρCNT , αnf =

knfρnf (cp)nf

,

knfkf

=(1− ϕ) + 2ϕ kCNT

kCNT−kfln

kCNT+kf2kf

(1− ϕ) + 2ϕ kfkCNT−kf

lnkCNT+kf

2kf

, νnf =µnf

ρnf, (2.9)

Further Table 2.1 presents CNTs liquid properties. Considering transformations:

Table 2.1: Outcomes for CNTs liquid [99].

Physical properties Base liquid NanoparticlesWater SWCNT MWCNT

ρ (kg/m3) 997 2600 1600cp (J/kgK) 4179 425 796k (W/mK) 0.613 6600 3000

η =

√U0

νl

(r2 −R2

2R

), ψ =

√U0νfxRf (η) , w =

U0z

lf ′ (η) ,

θ (η) =T − Tm∗

T∞ − Tm∗, u = −

√νU0

l

R

rf (η) , ζ(η) =

a

a0, h(η) =

b

a0, (2.10)

13

equation (2.3) is satisfied automatically and Eqs. (2.4− 2.8) become

(1

(1− ϕ)5/2(1− ϕ+ ϕρCNT

ρf)

)((1 + 2γη)f ′′′ + 2γf ′′) + ff ′′ − (f ′)2

−

(k1

(1− ϕ)5/2(1− ϕ+ ϕρCNT

ρf)

)f ′ = 0, (2.11) knf/kf

(1− ϕ+ ϕ (ρcp)CNT

(ρcp)f)

((1 + 2γη)θ′′) + 2γθ′) + Pr fθ′ = 0, (2.12)

(1 + 2γη) ζ ′′ + 2γζ ′ + Scfζ ′ − ScKζ(1− ζ)2 = 0, (2.13)

f ′(0) = 1, θ (0) = 0,

(knfkf

Mθ′(0) + (1− ϕ+ ϕρCNT

ρf) Pr f(0)

)= 0,

ζ′(0) = Ksζ(0), f ′(∞) = 0, θ (∞) = 1, ζ(∞) = 1. (2.14)

The involved variables are given below:

γ =

(νl

U0R2

) 12

, Pr =v

αf

, K =kra

20l

U0

, k1 =νfk∗U0

Sc =ν

DA

, Ks =ksDA

√νf l

U0

, M =Cp(T∞ − Tm∗)

λ3 + Cs(Tm∗ − T0). (2.15)

The local Nusselt number and skin friction;

Cf =τwρfw2

e

, Nuz =zqw

kf (Tw − T∞),

τw = µnf

(∂u

∂r

)r=R

, qw = −κnf(∂T

∂r

)r=R

. (2.16)

14

In dimensionless form these can be written as

NuzRe−1/2z = −knf

kfθ′(0), CfRe

1/2z =

1

(1− ϕ)5/2f ′′(0), (2.17)

where Reynolds number is Rez = wez/ν.

2.2 Homotopic results

The linear operators and initial guesses are defined by

f0(η) = (1− exp(−η)−knf

kfM

(1− ϕ+ ϕρCNT

ρf) Pr

), θ0(η) = (1− exp(−η)),

ζ0(η) = (1− 1

2exp(−Ksη)), (2.18)

Lf (f) =d3f

dη3− df

dη, Lθ (θ) =

d2θ

dη2− θ, Lζ (ζ) =

d2ζ

dη2− ζ, (2.19)

Lf [A1 + A2 exp(η) + A3 exp(−η)] = 0, (2.20)

Lθ [A4 exp(η) + A5 exp(−η)] = 0, (2.21)

Lζ [A6 exp(η) + A7 exp(−η)] = 0. (2.22)

Convergence domain of series outcomes depend upon auxilary varible ~. Figs. (2.2 −

2.3) presents ~-curves. Domain of convergence for auxiliary variables ~f , ~θ and ~ζ via

SWCNTs liquid is −1.6 ≤ ~f ≤ −0.1, −0.5 ≤ ~θ ≤ −0.3 and −1.3 ≤ ~ζ ≤ −0.3

while via MWCNTs liquid it is scaled as −1.15 ≤ ~f ≤ −0.25, −0.5 ≤ ~θ ≤ −0.3 and

−0.4 ≤ ~ζ ≤ −0.3.

15

2.3 Discussion

For outlines (i.e. f ′(η), θ(η), ζ(η)), the graphical results are discussed here. The

values of dimensionless parameters for results are k1 = 0.1, M = 0.1, γ = 0.1, ϕ = 0.1,

Pr = 6.2, K = 0.7, Ks = 0.9 and Sc = 1.2. These variables are fixed except the

variable mentioned in Figures. Fig. 2.4 displays influence of ϕ on the velocity outline. It

is noted that velocity of CNTs liquid enhances via ϕ. Fig. 2.5 presents the velocity outline

via k1. Here velocity outlines decreases via larger k1. In fact the resistive force enhances

via permeable mechanism which declines the CNTs liquid velocity. Velocity outline via

M is presented via Fig. 2.6. The liquid flow rises via larger M . Larger the variable

M correspond shift heated liquid to cold surface which leads the fluid velocity enhances.

Velocity outline via variable γ is displayed in Fig. 2.7. Near the surface the flow decays

and rises away the surface. R declines via γ. The contact area of liquid become less and

the flow rises up. MWCNTs liquid shows higher flow rate than SWCNTs liquid.

Melting variable M consequences via temperature outline is shown in Fig. 2.8. Ther-

mal boundary layer thickness enhances for melting parameter M in both SWCNT and

MWCNT cases. Fig. 2.9 addresses the temperature outline via γ. Temperature outline

reduces close to surface and it rises away the surface.

Curvature variable γ characteristics on concentration ζ(η) outline is addressed via Fig.

2.10. Concentration outline ζ(η) enhances adjacent to cylinder and opposite action is

noticed aside cylinder. Influence of M on concentration outline is plotted via Fig. 2.11.

LargerM declines the concentration outline. The concentration reports opposite responses

via variables K and Ks (see Figs. 2.12-2.13) respectively. Concentration outline via Sc

is presented in Fig. 2.14. Concentration outline ζ(η) enhances via large values of variable

16

Sc.

Fig. 2.15 is sketched via ϕ and k1 on CfRe1/2z . The CfRe

1/2z outline enhances larger

k1 and ϕ. Fig. 2.16 is presented via curvature variable γ and permeability variable k1 on

CfRe1/2z . Large k1 and γ enhance outcomes of CfRe

1/2z . MWCNTs liquid noted higher

skin friction than SWCNTs liquid. Fig. 2.17 addresses NuzRe−1/2z outline plotted via

variables ϕ and k1. NuzRe−1/2z outline increases via larger ϕ and opposite trend is noted

via k1. NuzRe−1/2z via MWCNTs liquid is higher than SWCNTs liquid. Fig. 2.18 reveals

the influence of M and γ variables on NuzRe−1/2z outline. Here NuzRe

−1/2z increases

via larger M and it decreases via γ. Nusselt number is noted higher via SWCNTs liquid

than MWCNTs liquid. SWCNTs liquid needs 30th and MWCNTs liquid needs 20th for

convergence (see Table 2.2).

17

hθ

θ’(0

)

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.10.75

0.8

0.85

0.9

0.95

SWCNT-Water

MWCNT-Water

Fig. 2.2: ~-curve for θ(η).

hf , hζ

f’’(

0),

ζ’(0

)

-3 -2 -1 0 1 2-1.5

-1

-0.5

0

0.5

1

1.5

SWCNT-WaterMWCNT-WaterSWCNT-WaterMWCNT-Water

ζ ’(0)

f ’’ (0)

Fig. 2.3: ~-curves for f(η) and ζ(η).

18

η

f’(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ = 0.0, 0.3, 0.4

SWCNT-Water

MWCNT-Water

Fig. 2.4: Plots via ϕ for f ′(η)

η

f’(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k1 = 0.0, 0.5, 1

SWCNT-Water

MWCNT-Water

Fig. 2.5: Plots via k1 for f ′(η).

19

η

f’(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 0.0, 0.5, 0.7

SWCNT-Water

MWCNT-Water

Fig. 2.6: Plots via M for f ′(η).

η

f’(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

Fig. 2.7: Plots via γ for f ′(η).

20

η

θ(η

)

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 0.0, 0.5, 0.7

SWCNT-Water

MWCNT-Water

Fig. 2.8: Plots via M for θ(η).

η

θ(η

)

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

γ = 0.0, 0.5, 0.9

SWCNT-Water

MWCNT-Water

Fig. 2.9: Plots via γ for θ(η).

21

η

ζ(η

)

0 1 2 3 4 5 6

0.4

0.5

0.6

0.7

0.8

0.9

1

γ = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

Fig. 2.10: Plots via γ for ζ(η).

η

ζ(η

)

0.9 1 1.1

0.625

0.63

0.635

0.64

0.645

0.65

η

ζ(η

)

0 1 2 3 4 5 6

0.4

0.5

0.6

0.7

0.8

0.9SWCNT-Water

MWCNT-Water

η

ζ(η

)

0.8 0.9

0.595

0.6

0.605

0.61

0.615

0.62

0.625

M = 0.0, 0.5, 0.7

Fig. 2.11: Plots via M for ζ(η).

22

η

ζ(η

)

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

Fig. 2.12: Plots via K for ζ(η).

η

ζ(η

)

0 1 2 3 4 5 60.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ks = 0.9, 1.2, 1.5

SWCNT-Water

MWCNT-Water

Fig. 2.13: Plots via Ks for ζ(η).

23

η

ζ(η

)

0 1 2 3 4 5 6

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sc = 0.0, 0.7, 1.7

SWCNT-Water

MWCNT-Water

Fig. 2.14: Plots via Sc for ζ(η).

φ

CfR

ez1

/2

0 0.025 0.05 0.075 0.1

-3

-2.5

-2

-1.5

k1 = 0.1, 0.2, 0.3

SWCNT-Water

MWCNT-Water

Fig. 2.15: Plots via ϕ and k1 for skin friction coefficient.

24

k1

CfR

e z1/2

0 1 2 3 4-2.2

-2.1

-2

-1.9

-1.8

-1.7

-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1

γ = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

Fig. 2.16: Plots via γ and k1 for skin friction coefficient.

k1

Nu z

Re z-1

/2

0 1 2 3 4

-1.75

-1.7

-1.65

-1.6

-1.55

-1.5

-1.45

-1.4

-1.35

-1.3

-1.25

-1.2

-1.15

φ= 0.0, 0.01, 0.05

SWCNT-Water

MWCNT-Water

Fig. 2.17: Plots via ϕ and k1 for Nussetl number.

25

M

Nu

zR

e z-1/2

0 0.025 0.05 0.075 0.1

-1.6

-1.55

-1.5

-1.45

-1.4

-1.35

-1.3

-1.25

-1.2

-1.15

γ = 0.1, 0.3, 0.5

SWCNT-Water

MWCNT-Water

Fig. 2.18: Plots via M and γ for Nussetl number.

Table 2.2: Convergence of equations via γ = 0.2, k1 = 0.1, M = 0.1,Ks = 1.2,K = 0.4,ϕ = 0.1 and Sc = 1.5.

SWCNTs MWCNTsEstimations order −f ′′(0) θ′(0) ζ ′(0) −f ′′(0) θ′(0) ζ ′(0)1 1.0133 0.78929 0.39659 1.0024 0.80981 0.0444415 1.0222 0.78031 0.36318 1.0037 0.79380 0.04116610 1.0297 0.78802 0.34383 1.0057 0.79423 0.03939313 1.0353 0.78144 0.33298 1.0074 0.79919 0.03846720 1.0397 0.78737 0.32699 1.0074 0.80663 0.03846725 1.0435 0.80473 0.32699 1.0074 0.80663 0.03846730 1.0435 0.80473 0.32699 1.0074 0.80663 0.038467

2.4 Main outcomes

Melting and diffusion species effects in CNTs liquid is presented. Key outcomes are

• Velocity outline becomes higher in MWCNTs liquid than SWCNTs liquid via γ and

ϕ.

• Melting variable enhances thermal and velocity outlines. SWCNTs corresponds to

maximum temperature than MWCNTs.

26

• Heterogeneous variable K declines the concentration outline.

• Temperature outline and concentration outline disclose decreasing behavior close to

the surface and increasing behavior beyond the cylinder

• Skin friction coefficient and local Nusselt number are increasing functions of volume

fraction ϕ and permeability parameter k1.

• It is found that high heat transfer and low thermal resistance via MWCNTs liquid

when compared with other CNTs liquids. .

27

Chapter 3

Diffusion species in convective CNTs flow through a permeable space

CNTs liquid flow via cylinder is addressed in this chapter. Diffusion species of reac-

tions and convective conditions are accounted for liquid flow. SWCNTs liquid and MWC-

NTs liquid are treated the nanofluids. Outcomes for equations are developed via OHAM.

Graphical outcomes are interpreted via variables.

3.1 Formulation

CNTs flow via stretched cylinder through permeable mechanism is considered. Dif-

fusion species and convective conditions are accounted. The hot liquid via Tf libeled the

temperature at surface of cylinder. The coordinates chosen are shown in Fig. 3.1. The

equations following boundary layer approximations are

28

Fig. 3.1: Geometry of problem

∂ (ru)

∂r+∂ (rw)

∂z= 0, (3.1)

w∂w

∂z+u

∂w

∂r= νnf

(∂2w

∂r2+

1

r

∂w

∂r

)− νnf

k∗w, (3.2)

w∂T

∂z+u

∂T

∂r= αnf

(1

r

∂T

∂r+∂2T

∂r2

), (3.3)

w∂a

∂z+u

∂a

∂r= DA

(∂2a

∂r2+

1

r

∂a

∂r

)− krab

2,

u∂b

∂r+w

∂b

∂z= DB

(∂2b

∂r2+

1

r

∂b

∂r

)+ krab

2, (3.4)

with

at r = R, u = 0, w = we =U0z

l, DB

∂b

∂r= −ksa,

− knf

(∂T

∂r

)= hf (Tf − T ) ,

as r → ∞, w → 0, a→ a0, T → T∞, b→ 0. (3.5)

29

Using Eq. 2.7 and Table 2.1 [96] and transformation:

η =

√U0

νl

(r2 −R2

2R

), ψ =

√U0νfzRf (η) , w =

U0z

lf ′ (η) ,

u =−√νU0

l

R

rf (η) , θ (η) =

T − T∞Tf − T∞

, ζ(η) =a

a0, h(η) =

b

a0, (3.6)

Eq. (3.1) and equations (3.2− 3.6) yield

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

) ((1 + 2γη) f ′′′ + 2γf ′′) + ff ′′ − (f ′)

2

−

k1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

) f ′ = 0, (3.7)

knf/kf(1− ϕ+ ϕ

(ρcp)CNT

(ρcp)f

) ((1 + 2γη) θ′′ + 2γθ′) + Pr fθ′ = 0, (3.8)

1

Sc((1 + 2γη)ζ ′′ + γζ ′) + fζ ′ −Kζh2 = 0, (3.9)

δ1Sc

((1 + 2γη)h′′ + γh′) + fh′ +Kζh2 = 0, (3.10)

f(0) = 0, f ′(0) = 1, θ′(0) = − kf

knfBi (1− θ(0)) , f ′(∞) = 0, θ (∞) = 0,

ζ ′(0) = Ksζ(0), ζ(η) → 1, δ1h′(0) = −Ksζ(0) h(η) → 0 as η → ∞,

(3.11)

The parameters appearing in Eqs. (3.8− 3.12) are defined below:

γ =

(νf l

U0R2

) 12

, P r =µf (cp)fkf

, K =kra

20l

U0

, k1 =νfk∗U0

Sc =νfDA

, Ks =ksDA

√νf l

U0

, Bi =hfkf

√νf l

U0

, δ1 =DB

DA

. (3.12)

30

For same diffusion coefficients DA and DB, one has

ζ(η) + h(η) = 1.

Eqs. (3.10), (3.11) and (3.12) yield

(1 + 2γη)ζ ′′ + γζ ′ + Scfζ ′ − ScKζ(1− ζ)2 = 0, (3.13)

ζ ′(0) = Ksζ(0), ζ(η) → 1 as η → ∞. (3.14)

The Skin friction and Nusselt number;

Cf =τwρfw2

e

, Nuz =zqw

kf (Tw − T∞),

τw =µnf

(∂u

∂r

)r=R

, qw = −κnf(∂T

∂r

)r=R

. (3.15)

Dimensionless variables finally yield

CfRe1/2z =

1

(1− ϕ)2.5f ′′(0), NuzRe

−1/2z = −knf

kfθ′(0), (3.16)

where Rez = wez/ν.

31

3.2 OHAM outcomes

The linear operator and initial guesses are defined by

f0 = (1− e−η), θ0 =Bi

(knf

kf+Bi)

e−η, ζ0 = (1− 1

2e−Ksη), (3.17)

Lf (f) =d3f

dη3− df

dη, Lθ (θ) =

d2θ

dη2− θ, Lζ (ζ) =

d2ζ

dη2− ζ. (3.18)

For optimal results of variables hf , hθ and hζ , we defined the average squared residual

errors at kth order

εfk (hf ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη

]2, (3.19)

εθk (hf , hθ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη ,

k∑i=0

(θi)η=jπη

]2, (3.20)

εζk (hf , hζ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη ,

k∑i=0

(ζi)η=jπη

]2, (3.21)

εt = εfk + εθk + εζk. (3.22)

Where εt denotes the total square residual error. The obtained results of optimal variables

via SWCNTs liquid as hf = −1.43306, hθ = −0.310634 and hζ = −0.620428 and the

total residual error is εt = 1.65 × 10−4 while optimal results via MWCNTs liquid are

hf = −0.657264, hθ = −0.345468, hζ = −0.59548 and the error is εt = 2.06× 10−4.

3.3 Discussion

Main focus here is to address velocity, temperature and concentration outline via em-

bedded variables for CNTs liquid. The values of dimensionless variables for solution are

32

k1 = 0.1, γ = 0.1, ϕ = 0.1, β = 0.1, Da = 0.1, Sc = 1.2, Ks = 0.4, K = 0.8,

Pr = 6.2. These variables are treated as constant except the variable shown in the Figs.

Fig. 3.2 presents the velocity outline via k1. The CNTs flow decline via larger k1. The

resistive force via permeable medium declines the CNTs flow. Fig. 3.3 addresses the ve-

locity outline via γ. Velocity outline decreases close to the cylinder and enhances away

the cylinder via larger γ. Fig. 3.4 presents the plots of velocity outline via ϕ. The velocity

outlines enhance via ϕ. It is noticed that velocity outlines for MWCNTs liquid is higher

than SWCNTs liquid. The thermal outline viaBi is addressed in Fig. 3.5. SWCNTs liquid

noted higher temperature outline than MWCNTs liquid. Fig. 3.6 shows γ effects on θ. The

temperature outline decays near the cylinder and enhances away from cylinder via larger

γ. Fig. 3.7 reveals the plots via the variable γ for concentration outline. Concentration

outline becomes higher via larger γ. The concentration outline via larger K and Ks are

demonstrated via Figs. 3.8 − 3.9 respectively. Concentration outline declines via K and

enhances via Ks. The plots of Sc via concentration outline is displayed in Fig. 3.10. Con-

centration outline enhances via larger Sc. Skin friction outline via γ and k1 is displayed in

Fig. 3.11. The outline enhances via variables k1 and γ. Skin friction in SWCNTs liquid

is noted higher in magnitude than that MWCNTs liquid. Nusselt number outline via ϕ and

γ is presented in Fig. 3.12. Magnitude of Nusselt number enhances via ϕ and declines

via larger γ. Nusselt number outcomes in SWCNTs liquid is noticed larger. Figs. 3.11

and 3.12 present the validation OHAM outline and numerical outline via skin friction. The

oulines show the results meet.

Table 3.1 represents the average square residual errors. As the order of approximation in-

creases the error declines. Table 3.2 addresses the convergence of outcomes. It is observed

that momentum, energy and concentration outlines need 13th order via SWCNTs. The

33

momentum and energy equations converge at 10th order via MWCNTs and concentration

results become same at 20th order via MWCNTs. Table 3.3 represents results of previous

findings and present finding via f ′′(0) at ϕ = 0.0, γ = 0.0. Outcomes found in an excellent

agreement.

34

η

f’(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k1 = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

Fig. 3.2: Plots via k1 for f ′(η).

η

f’(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

Fig. 3.3: Plots via γ for f ′(η).

35

η

f’(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ= 0.0, 0.2, 0.4

SWCNT-Water

MWCNT-Water

Fig. 3.4: Plots via ϕ for f ′(η).

η

θ(η

)

0 1 2 3 4 5 6

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Bi = 0.3, 0.5, 0.7

SWCNT-Water

MWCNT-Water

Fig. 3.5: Plots via Bi for θ(η).

36

η

θ(η

)

0 1 2 3 4 5 6

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

γ = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

Fig. 3.6: Plots via γ for θ(η).

η

ζ(η

)

0 1 2 3 4 5 6

0.4

0.5

0.6

0.7

0.8

0.9

1

SWCNT-Water

MWCNT-Water

γ = 0.0, 0.5, 1.0

Fig. 3.7: Plots via γ for ζ(η).

37

η

ζ(η

)

0 1 2 3 4 5 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

Fig. 3.8: Plots via K for ζ(η).

η

ζ(η

)

0 5 10 15 200.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ks = 0.7, 0.9, 1.2

SWCNT-Water

MWCNT-Water

Fig. 3.9: Plots via Ks for ζ(η).

38

η

ζ(η

)

0 2 4 6 80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sc = 0.5, 0.9, 1.7

SWCNT-Water

MWCNT-Water

Fig. 3.10: Plots via Sc for ζ(η).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−3

−2.5

−2

−1.5

−1

−0.5

γ

CfR

e1

/2z

OHAMOHAMOHAMOHAMOHAMOHAMBVP4cBVP4cBVP4cBVP4cBVP4cBVP4c

MWCNT+Water

SWCNT−Water

k1 = 0.0, 0.5, 0.9

Fig. 3.11: Plots for skin friction via γ and k1.

39

0.3 0.32 0.34 0.36 0.38 0.4 0.420.009

0.01

0.011

0.012

0.013

0.014

0.015

γ

Nu

zR

e−

1/2

OHAMOHAMOHAMOHAMOHAMOHAMBVP4cBVP4cBVP4cBVP4cBVP4cBVP4c

φ = 0.1, 0.7, 1.2

MWCNT+Water

SWCNT+Water

Fig. 3.12: Plots for Nusselt number ϕ via γ.

40

Table 3.1: Numerical results of individual residual errors via SWCNTs liquid and MWC-NTs liquid at different order with γ = 0.1, k1 = 0.1, ϕ = 0.1, Bi = 0.1, K = 0.4,Ks = 1.2 and Sc = 1.5.

SWCNTs MWCNTsk εfk εθk εζk εfk εθk εζk2 5.44×10−7 8.86×10−6 1.56×10−4 1.06×10−6 1.88×10−5 1.86×10−4

4 7.24×10−8 6.30×10−7 1.63×10−5 2.22×10−9 2.36×10−6 1.95×10−5

8 1.28×10−9 5.79×10−9 1.72×10−6 1.27×10−14 6.32×10−8 2.04×10−6

10 1.71×10−10 4.53×10−10 1.70×10−7 3.46×10−17 9.04×10−9 2.00×10−7

12 2.28×10−11 3.17×10−11 1.64×10−8 1.00×10−19 1.04×10−9 1.91×10−8

14 3.03×10−12 5.98×10−12 1.55×10−9 3.11×10−22 1.57×10−10 1.80×10−9

16 4.05×10−13 6.40×10−13 1.45×10−10 1.00×10−34 1.33×10−11 1.66×10−11

Table 3.2: Convergence via γ = 0.2, k1 = 0.1, Ks = 0.1, K = 0.2, Bi = 0.2, ϕ = 0.2and Sc = 1.3 for the series solutions.

SWCNTs MWCNTsEstimations order −f ′′(0) −θ′(0) ζ ′(0) −f ′′(0) −θ′(0) ζ ′(0)

1 1.01871 0.0635 0.04734 1.00791 0.0674 0.047345 1.03693 0.0633 0.04412 1.01313 0.0671 0.0441210 1.04323 0.0632 0.04265 1.01313 0.0671 0.0426613 1.04544 0.0632 0.04173 1.01313 0.0671 0.0406720 1.04544 0.0632 0.04173 1.01313 0.0671 0.0396325 1.04544 0.0632 0.04173 1.01313 0.0671 0.995930 1.04544 0.0632 0.04173 1.01313 0.0671 0.9959

Table 3.3: Comparison of f ′′(0) via k1 when γ = 0, ϕ = 0 [97].

k1 [97] PR0.5 1.22474487 1.22471 1.41421356 1.4142

1.5 1.58113883 1.58112 1.73205081 1.73205 2.44948974 2.4494

41

3.4 Main findings

Convective CNTs flow with diffusion species via permeable medium is explored. Main

results are

• Velocity and temperature outline enhances via larger γ.

• Homogeneous variable K declines the concentration outline.

• Skin friction enhances via larger permeability variable k1.

• Large ϕ correspond to high magnitude of Nusselt number outcome and low magni-

tude is noted via large γ.

42

Chapter 4

Computational study for CNTs nanofluid with renovated

Hamilton-Crosser and Xue models past a stretching cylinder

Present chapter explores CNTs flow past a cylinder and flat sheet. Thermal conduc-

tivity of CNTs nanofluid is inspected under renovated Hamilton-Crosser and Xue models.

Thermodynamic processes of reactant and autocatalyst analyze the impact of temperature

phase changes like evaporation or convection. Nanoliquid is composed via SWCNTs liq-

uid and MWCNTs liquid. The solution of problem is obtained by using implicit finite dif-

ference method (Keller-Box). Influence of physically involved parameters are illustrated

through graphs. The velocity outline and thermal outline via cylinder is noted higher than

flat sheet.

4.1 Constructions

Flow of CNTs based liquid is investigated via a flat sheet and a stretching cylinder.

An incompressible liquid fills non-Darcy permeable medium. Stretching cylinder and flat

sheet is made for CNTs (SWCNTs, MWCNTs) nanoliquids. The flow of nanofluid is due

to stretching cylinder. Stretching outline is developed via two equal and opposite forces.

43

Fig. 4.1: Physical model.

It is assumed that Tw > T∞. The thermal conductivity of CNTs nanofluid is first analyzed

via renovated H-C model and then equate with Xue model. The impact of diffusion species

is investigated in the CNTs flow for thermodynamic processes. The coordinates selected

is shown in Fig. 4.1. Applying the standard boundary layer approximations the equations

become:

∂ (ru)

∂r+∂ (rw)

∂z= 0, (4.1)

w∂w

∂z+u

∂w

∂r= νnf

(∂2w

∂r2+

1

r

∂w

∂r

)− νnf

ϕ⋆

k∗w − cbϕ

⋆

√k∗w2, (4.2)

w∂T

∂z+u

∂T

∂r= αnf

(1

r

∂T

∂r+∂2T

∂r2

), (4.3)

u∂a

∂r+w

∂a

∂z= DA

(∂2a

∂r2+

1

r

∂a

∂r

)− krab

2,

u∂b

∂r+w

∂b

∂z= DB

(∂2b

∂r2+

1

r

∂b

∂r

)+ krab

2, (4.4)

44

The boundary conditions are defined bellow:

at r = R, w = we =

U0zl, u = 0, T = Tw, DB

∂b∂r

= −ksa,DA∂a∂r

= ksa,

as r → ∞, w → 0 a→ a0, b→ 0, T → T∞.

(4.5)

The model proposed by Xue (i.e. Eq. 2.9) and renovated Hamilton-Crosser model [98]

yields

knfkf

=kpe + kf (n− 1) + (n+ 1)(kpe − kf )(1 + C)ϕ

kpe + kf (n− 1)− (kpe − kf )(1 + C)ϕ,

kpe =1

π

∫ π

0

√k2pez sin

2 θ + k2pex cos2 θ1dθ1, kpex =

A1ϕkCNT +B1Cϕklr

A1ϕ+B1Cϕ,

kpez =kCNT + Cklr

1 + C, klr =

kCNTR1(1 + t1/R1 − kf/kCNT ) ln(1 + t1/R1)

tkf ln[(1 + t1/R1)kCNT/kf ],

A1 = − 2klr

kCNTklr, B1 =

R1

R1 + t1

(kCNT − klr

klr + kCNT

− 1

), C =

(R1 + t1)2 −R2

1

R21

,

n1 = 3ψ−g2 , ψ =2e(u)(1− e(u)2)1/6

e(u)(1− e(u)2 arcsin(e(u)))0.5, e(u) = (1− R2

1 + u

(L1/2)2 + u)0.5.

(4.6)

.

CNTs properties are presented in Table 2.1. Using the relations

η =

√U0

νl

(r2 −R2

2R

), w =

U0z

lf ′ (η) , ψ =

√U0νfzRf (η) ,

u =−√νU0

l

R

rf (η) , ζ(η) =

a

a0, h(η) =

b

a0, θ(η) =

T − T∞Tw − T∞

, (4.7)

45

equation 4.1 is satisfied trivially while Eqs. (4.2-4.6) are reduced to

(1

(1− ϕ)52 (1− ϕ+ ϕρCNT

ρf))((1 + 2γη)f ′′′ + 2γf ′′) + ff ′′ − (1 + β)(f ′)2

− (Da

(1− ϕ)52 (1− ϕ+ ϕρCNT

ρf))f ′ = 0, (4.8)

(knf/kf

(1− ϕ+ ϕ (p)CNT

(ρcp)f))((1 + 2γη)θ′′ + 2γθ′) + Pr fθ′ = 0, (4.9)

1

Sc((1 + 2γη)ζ ′′ + γζ ′) + fζ ′ −Kζh2 = 0, (4.10)

δ1Sc

((1 + 2γη)h′′ + γh′) + fh′ +Kζh2 = 0, (4.11)

with boundary conditions

f(0) = 0, f ′(0) = 1, f ′(∞) = 0, θ(0) = 1, ζ ′(0) = Ksζ(0),

θ(∞) → 0, ζ(∞) → 1, δ1h′(0) = −Ksh(0), h(∞) → 0,

(4.12)

The parameters are defined below:

γ =

(νf l

U0R2

) 12

, K =kra

20l

U0

, Da =ϕ⋆νfk∗we

,

β =cbϕ

⋆

√k⋆, Sc =

νfDA

, Ks =ksDA

√νf l

U0

, δ1 =DB

DA

. (4.13)

For same diffusion coefficients DA and DB [83], one has

ζ(η) + h(η) = 1.

46

Now Eqs. (4.11), (4.12) and (4.13) become

(1 + 2γη)ζ ′′ + γζ ′ + Scfζ ′ − ScKζ(1− ζ)2 = 0.

ζ ′(0) = Ksζ(0), ζ(η) → 1 as η → ∞. (4.14)

The skin friction and Nusselt number;

Cf =τwρfw2

e

, Nuz =zqw

kf (Tw − T∞),

τw =µnf

(∂u

∂r

)r=R

, qw = −κnf(∂T

∂r

)r=R

, (4.15)

or

CfRe1/2z =

1

(1− ϕ)2.5f ′′(0), NuzRe

−1/2z = −knf

kfθ′(0), (4.16)

where Rez = wez/ν is the Reynolds number.

4.2 Keller-box results

It [112] is an implicit unconditionally stable technique capable of solving variety of

different engineering problems. There are four main steps in Keller box method. i.e. (i)

reduce nth order to nth1st order equations. (ii) discretization (i.e. finite difference) (iii)

Quasilinearization (i.e. algebraic equations) (iv) Block-tridiagonal elimination.

47

4.2.1 Reduction of the nth order system to nth 1st order equations

Reduce Eqs.(4.7 − 4.13) into first order system. We introduce the following new de-

pendent variables u, v, w, g, t, p and q for momentum, energy and concentration equations

and obtain the following first order equations

t = g′, q = p′, u = f ′, v = u′, (4.17)

equations (4.11-4.17) are converted into forms:

(1

(1−ϕ)5/2(1−ϕ+ϕρCNT

ρf))[(1 + 2ηγ)v′ + 2γv] + fv − (1 + β)u2 −Dau, (4.18)

(knf/kf

(1− ϕ+ ϕ (ρcp)CNT

(ρcp)f))[(1 + 2γη)t′ + 2γt] + Pr tf = 0. (4.19)

1

Sc[(1 + 2Kη)q′ + 2Kq] + qf −Kp(1− p)2 = 0. (4.20)

4.2.2 Finite difference discretization

The Eqs.(4.19− 4.21) in difference form via x− η plane with net points as follows:

z0 = 0, zi = zi−1 + ki, i = 1, ...I, (4.21)

η0 = 0, ηi = ηj−1 + hj, j = 1, ...J. (4.22)

Here ki-hj the ∆z-∆η-layout respectively. Centralizing via (zi, ηj−1/2)

48

Fig. 4.2: Net ”Keller box” for different approximations.

f ij − f i

j−1

hj=uij + uij−1

2, (4.23)

uij − uij−1

hj=vij + vij−1

2, (4.24)

gij − gij−1

hj=tij + tij−1

2(4.25)

pij − pij−1

hj=qij + qij−1

2(4.26)

49

4.2.3 Quasilinearization of non-linear Keller algebraic equations

Now centering at (zi−1/2, ηj−1/2)

(1

(1−ϕ)2.5(1−ϕ+ϕρCNT

ρf))((1 + 2γη)(vij − vij−1) + 2γhvij−1/2) + hf i

j−1/2vij−1/2

− h(1 + β)(uij−1/2)2 − hDa(uij−1/2) = Ej−1/2, (4.27)

(knf/kf

(1− ϕ+ ϕ (ρcp)CNT

(ρcp)f))((1 + 2γη)(tij − tij−1) + 2γhtij−1/2)

+ Prhtij−1/2fij−1/2 = Lj−1/2, (4.28)

1

Sc((1 + 2γη)(gij − gij−1) + 2γhgij−1/2) + hgij−1/2f

ij−1/2

−Khpij−1/2(1− pij−1/2)2 =Mj−1/2, (4.29)

where

Ej−1/2 =−

1

(1−ϕ)2.5(1−ϕ+ϕ

ρCNTρf

) [(1 + 2γη)(vi−1

j − vi−1j−1) + 2γhvi−1

j−1/2]

+ hf i−1j−1/2v

i−1j−1/2 − h(1 + β)(ui−1

j−1/2)2 −

1

(1−ϕ)2.5(1−ϕ+ϕ

ρCNTρf

)hDa(ui−1

j−1/2),

(4.30)

Lj−1/2 =− ((knf/kf

(1− ϕ+ ϕ (ρcp)CNT

(ρcp)f))[(1 + 2γη)(ti−1

j − ti−1j−1) + 2γhti−1

j−1/2]

+ Prhti−1j−1/2f

i−1j−1/2), (4.31)

Mj−1/2 =− (1

Sc((1 + 2γη)(qi−1

j − qi−1j−1) + 2γhqi−1

j−1/2) + hqi−1j−1/2f

i−1j−1/2)

− hKpi−1j−1/2(1− pi−1

j−1/2)2, (4.32)

50

here the known variables are Ej−1/2, Lj−1/2 and Mj−1/2,

with

gi0 = 1, giJ = 0, f i0 = 0, ui0 = 1, uiJ = 0, qi0 = Kspi0 = 0, piJ = 1. (4.33)

As Eqs.(4.28 − 4.30) are nonlinear equations (algebraic). Linearized these equations via

Newton’s method.

4.2.4 The Block tridiagonal matrix

The linearized differential equation of system (Eq. 4.40−4.47) has a block tridiagonal

structure written by

[A1

] [C1

][B2

] [A2

] [C2

]·

·

·

·

·

· [Bj−1

] [Aj−1

] [Cj−1

][BJ

] [AJ

]

[δ∗1

][δ∗2

]·

·

·[δ∗j−1

][δ∗J

]

=

[r1

][r2

]·

·

·[rj−1

][rJ

]

,

(4.34)

that is [A

] [δ∗

]=

[r

]. (4.35)

51

The entries defined in Eq. (4.95) put into the forms:

[A1

]=

0 0 0 1 0 0 0

d 0 0 0 d 0 0

0 d 0 0 0 d 0

0 0 d 0 0 0 d

(a2)1 0 0 (a3)1 (a1)1 0 0

0 (b2)1 0 (b3)1 0 (b1)1 0

0 0 (c2)1 (c3)1 0 0 (c1)1

, d = −h12,

[Aj

]=

d 0 0 1 0 0 0

−1 0 0 0 d 0 0

0 −1 0 0 0 d 0

0 0 −1 0 0 0 d

(a6)j 0 0 (a3)j (a1)j 0 0

0 0 0 (b3)j 0 (b1)j 0

0 0 (c6)j (c3)j 0 0 (c1)j

, d = −h12, 2 ≤ j ≤ J,

[Bj

]=

0 0 0 −1 0 0 0

0 0 0 0 d 0 0

0 0 0 0 0 d 0

0 0 0 0 0 0 d

0 0 0 (a4)j (a2)j 0 0

0 0 0 (b4)j 0 (b2)j 0

0 0 0 (c4)j 0 0 (c2)j

, d = −h12, 2 ≤ j ≤ J,

52

[Cj

]=

d 0 0 0 0 0 0

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

(a5)j 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 (c5)j 0 0 0 0

, d = −h12, 2 ≤ j ≤ J − 1,

After linearization one has

[A][δ∗] = [r]. (4.36)

Eq.(4.104) is evaluated via LU method and [δ∗] is calculated from this block tridiagonal

matrix. The solution procedure is repeated until convergence criterion is achieved and

stopped when

δ∗v(i)0 ≤ ξ, (4.37)

where ξ = 0.001 is a small positive value.

4.2.5 Block-tridiagonal elimination of linear Keller algebraic equa-

tions

The linear system of Eqs. (4.47 − 4.54) can now be solved by the block-elimination

method. The block-tridiagonal structure is formed for linearized difference Eqs. (4.47 −

4.54). The mesh sensitivity analysis generates accurate outcomes. Some attempts in the

53

η-direction are done and a number of mesh points are obtained while in the x− direction

minimum mesh points are used. The boundary conditions are satisfied at η max.

4.3 Discussion

A comparative study of stretching cylinder and flat sheet is discussed through reno-

vated Hamilton-Crosser and Xue models. Here black and red lines show the profiles for

stretching cylinder in SWCNTs liquid and MWCNTs liquid respectively while solid and

broken blue lines show the profiles for flat sheet in SWCNTs liquid and MWCNTs liquid

respectively. The values of dimensionless parameters for numerical outcomes are ϕ = 0.1,

= 0.1, β = 0.1, γ = 0.1,Ks = 1, Sc = 1.5 and K = 0.2. These parameters are fixed

except the parameter as defined in figures. The outcomes of dimensionless parameters like

volume fraction, inverse Darcy number, local inertia parameter, curvature parameter etc

are disclosed for profiles (velocity, temperature and concentration). Further the outcomes

for skin friction and Nusselt number are also investigated through same parameters by Xue

and renovated H-C models.

Table 4.1 is prepared for skin friction for flat sheet γ = 0 and cylinder γ = 0.1 for

various values of ϕ, Da and β. It is observed that skin friction for CNTs nanoflud due to

stretching cylinder is larger when compared with flat sheet. Further the skin friction for

SWCNT-Water nanofluid is slightly dominated than MWCNT-Water. Table 4.3 represents

magnitude of Nusselt number against ϕ for cylinder and flat plate cases. It is observed that

the heat transfer rate due to stretching cylinder is higher when compared with flat plate.

Also the heat transfer in SWCNTs liquid is higher than MWCNTs liquid for both cylinder

and flat plate. Table 4.3 shows the skin friction and Nusselt number against curvature. Skin

friction via MWCNTs-liquid is higher when compared with SWCNTs-liquid. Magnitude

54

of heat transfer is higher for MWCNT-Water than SWCNT-Water for different values of

γ. Nusselt number and skin friction are increasing functions of curvature parameter. Table

4.4 shows comparison of skin friction in limiting case for different volume fraction.

Fig. 4.1 shows the net rectangle considered in the z−r plane, where i and j stand for the

sequence of numbers that represent the coordinate position. The functions (f, u, v, t, q) at

points (xi, ηj) are approximated by the net (fi, ui, vi, ti, qi) functions. Centered-difference

derivatives is used for the finite-difference approximations of Eqs. (4.33 − 4.35) at the

midpoint (xi, ηj−1/2) of the line segment (P1P2). The finite-difference forms of Eqs.

(4.33− 4.35) are calculated by centering about the midpoint (xi−1/2, ηj−1/2) of the rectan-

gle (P1P2P3P4). Fig. 4.2 represents the schematic diagram of the problem.

The plots of local inertia parameter β for velocity outline is addressed in Fig. 4.3.

Where γ = 0 (i.e. solid and broken blue lines) shows the plot for flat sheet whereas

γ = 0.1 (i.e. black and red lines) represents the plots in case of stretching cylinder. Larger

β leads to decline the velocity outline. The velocity profile in case of stretching cylinder is

higher when compared with flat sheet. Further velocity distribution for MWCNT is larger

than SWCNT. Similar behavior is seen for inverse Darcy number Da as shown in Fig.

4.4. The plots of volume fraction ϕ for velocity outline is sketched in Fig. 4.5. Larger

ϕ correspond to higher velocity profile for both flat sheet and stretching cylinder. The

velocity in case of stretching cylinder is higher when compared with flat sheet. It is also

noticed that the velocity profile for MWCNT is dominated than SWCNT. The velocity

distribution via curvature parameter γ is plotted in Fig. 4.6. Curvature parameter is an

increasing function of velocity profile. Physically it means that larger values of γ tend to

reduce the radius of cylinder. As a result the resistance for the flowing fluid becomes less

and consequently the velocity profile enhances.

55

Fig. 4.7 depicts the plots of ϕ via temperature profile for flat sheet and cylinder. Tem-

perature outline is enhanced for ϕ increases. The temperature for stretching cylinder is

prominent than flat sheet. Temperature field enhances when curvature parameter is in-

creased (see Fig. 4.8). Fig. 4.9 includes the plots of ϕ for concentration profile. The

concentration distribution is decreasing function of volume fraction. It is observed that the

concentration field in case of flat sheet is higher when compared with stretching cylinder.

Further the concentration in SWCNTs liquid is higher than MWCNTs liquid.

Fig. 4.10 discloses the curves of K via concentration outline against the flat plate

and stretching cylinder. The concentration outline via larger K declines. Concentration

via flat sheet is higher than stretching cylinder. The plots Sc via concentration outline is

addressed in Fig. 4.11. Concentration improves via larger Schmidt number. Concentration

outline enhances via flat sheet than cylindrical case. Skin friction coefficient via curvature

parameter and volume fraction is included in Fig. 4.12. It is revealed from Fig. that the

skin friction is enhanced via γ and ϕ. The comparison of renovated H-C model and Xue

model is presented in Fig. 4.13 for Nusselt number against ϕ and γ. Heat transfer via

renovated H-C model is higher than Xue model. Moreover, heat transfer rate via SWCNT

is higher than MWCNT.

56

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f ’ ( η

)

30 35 40 45 500.05

0.1

0.15

0.2

0.25

0.3

η

f’(η

)

β = 0.1, 0.7, 1.2

γ = 0.0

SWCNT

MWCNT

γ = 0.1

SWCNT

MWCNT

Fig. 4.3: Plots via β for f ′(η).

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f ’ ( η

)

25 30 35 40 450.05

0.1

0.15

0.2

0.25

0.3

η

f’(η

)

Da = 0.1, 0.7, 1.2

γ = 0.0

SWCNT

MWCNT

γ = 0.1

SWCNT

MWCNT

Fig. 4.4: Plots via Da for f ′(η).

57

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f ’ (η

)

12 14 16 18 20

0.1

0.15

0.2

0.25

0.3

0.35

η

f’ (η

)

φ = 0.1, 0.5, 0.7

γ = 0.1

SWCNT

MWCNT

γ = 0.0

SWCNT

MWCNT

Fig. 4.5: Plots via ϕ for f ′(η).

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f ’ ( η

)

15 20 25 30 35

0.05

0.1

0.15

0.2

0.25

0.3

η

f ’ ( η

)

γ = 0.1, 0.5, 0.9

SWCNT

MWCNT

Fig. 4.6: Plots via γ for f ′(η).

58

0 50 100 150 200 250 300 350 400−0.2

0

0.2

0.4

0.6

0.8

1

η

θ (

η )

110 120 130 140

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

η

θ(η

)

φ = 0.1, 0.5 , 0.9

γ = 0.0

SWCNT

MWCNT

γ = 0.1

SWCNT

MWCNT

Fig. 4.7: Plots via ϕ for θ(η).

0 50 100 150 200 250 300 350 400−0.2

0

0.2

0.4

0.6

0.8

1

1.2

η

θ (

η )

100 110 120 130

0

0.1

0.2

0.3

η

θ (

η)

γ = 0.1, 0.5, 0.9

Fig. 4.8: Plots via γ for θ(η).

59

0 50 100 150 200 2500.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

η

ζ (

η )

55 60 65 70 750.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

η

ζ(η

)

γ = 0.0

SWCNT

MWCNT

γ = 0.1

SWCNT

MWCNT

φ = 0.1, 0.7, 0.9

Fig. 4.9: Plots via ϕ for ζ(η).

0 50 100 150 200 2500.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

η

ζ (

η )

50 55 60 65 70

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

η

ζ (

η )

γ = 0.0

SWCNT

MWCNT

γ = 0.1

SWCNT

MWCNT

K = 0.1, 0.7, 1.5

Fig. 4.10: Plots via K for ζ(η).

60

0 50 100 150 200 2500.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

η

ζ (

η )

30 35 40 45 500.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

η

ζ (

η)

γ = 0.0

SWCNT

MWCNT

γ = 0.1

SWCNT

MWCNT

Sc = 0.7, 1, 1.2

Fig. 4.11: Plots via Sc for ζ(η).

γ

Re x0.

5C

fx

0.2 0.4 0.61

1.1

1.2

1.3

1.4

φ= 0.01

φ= 0.02

φ= 0.03

Fig. 4.12: Plots for skin friction via γ and ϕ.

61

γ

Re x-0

.5N

u x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

φ= 0.01φ= 0.01φ= 0.02φ= 0.02φ= 0.03φ= 0.03φ= 0.01φ= 0.01φ= 0.02φ= 0.02φ= 0.03φ= 0.03

renovated

Hamilton-Crosser model

Xue model

Fig. 4.13: Plots for Nusselt number via γ and ϕ.

Table 4.1: Skin friction for various values of ϕ, β, Da.

γ = 0.1 γ = 0ϕ β Da SWCNT MWCNT SWCNT MWCNT

0.1 0.1 0.1 1.35529971987438 1.35529905571849 1.30134883134501 1.3013488313450120.2 1.81934825675556 1.81934327416910 1.74692810742171 1.7469281074217110.3 2.54035930686713 2.54034922325491 2.43924205986611 2.4392420598661100.1 0.1 0.1 1.35529971987438 1.35529905571849 1.30134883134501 1.301348831345012

0.5 1.355318861136590 1.355311953236433 1.301348831345012 1.3013488313450120.1 0.1 0.1 1.35529971987438 1.35529905571849 1.30134883134501 1.301348831345012

0.5 1.355318861136590 1.355311953236433 1.301348831345012 1.301348831345012

62

Table 4.2: Nusselt number for ϕ in case of SWCNTs liquid and MWCNTs liquid.

γ = 0.1 γ = 0ϕ SWCNT MWCNT SWCNT MWCNT

0.01 4.0752 4.0435 2.3606 2.34080.02 6.2169 6.1562 3.7790 3.73680.03 8.2393 8.1512 5.2582 5.1908

Table 4.3: Skin friction and Nusselt number for CNTs liquid via various values of curvatureparameter γ.

Skin friction Nusselt numberγ SWCNT MWCNT SWCNT MWCNT

0.1 1.062788977512708 1.067962826997629 4.0435 4.07520.5 1.211977957614859 1.239231099079156 4.2672 4.30000.9 1.356230833176705 1.406072161622822 4.3189 4.3486

Table 4.4: Validation of skin friction [99] via Da = 0 and β = 0 for ϕ.

ϕ CNTs [99] PR0.01 SWCNT 0.33894 0.33890.01 MWCNT 0.33727 0.33720.1 SWCNT 0.40811 0.40810.1 MWCNT 0.39008 0.39000.2 SWCNT 0.50452 0.50450.2 MWCNT 0.46466 0.4646

63

4.4 Main findings

Computational study for CNTs nanofluid using renovated Hamilton-Crosser and Xue

models over a flat sheet and cylinder is addressed. The major outcomes are:

• Volume fraction corresponds to an enhancement in fluid flow, thermal boundary

layer thickness and magnitude of heat transfer coefficient for flat sheet and stretching

cylinder while opposite trend has been noted for concentration profile. The temper-

ature for cylindrical case is higher when compared with flat sheet. Moreover the

concentration is higher for flat sheet when compared with stretching cylinder.

• Curvature parameter is increasing function of temperature and velocity. The magni-

tude of heat transfer also enhances for larger γ.

• Homogeneous reaction declines the concentration while Schmidt number enhances

the concentration. The concentration outline via flat sheet is noted larger than via

stretching cylinder.

• Larger skin friction is noted for ϕ, β, Da and γ. Skin friction coefficient for cylinder

is higher than flat sheet. Skin friction for SWCNT-liquid is noted slightly larger than

MWCNTs-liquid.

• Heat transfer improves via renovated H-C model when compared with Xue model.

Higher heat transfer rate is noted via γ and ϕ. Higher magnitude of heat transfer is

observed for SWCNTs-liquid when compared with MWCNTs-liquid.

64

Chapter 5

Stagnation point in CNTs flow

Stagnation point CNTs flow of viscous nanoliquid induced by nonlinear stretched sheet

of deformable thickness is addressed. The results for SWCNTs liqud and MWCNT liquid

are obtained and matched. Kerosene oil and water are addressed as the base liquids. Salient

features of permeable mechanism, melting heat and diffusion reactions are analyzed. Op-

timal homotopy method (OHAM) is utilized for convergent series solutions development.

Residual errors are presented and examined. Local Nusselt number and skin friction are

highlighted via variables. Velocity is noticed more via ϕ in case of kerosene oil CNTs liq-

uid than water CNTs liquid. CfRe12z declines via larger ratio parameter. However CfRe

12z

enhances via larger volume friction. The flow accelerates for MWCNT than SWCNT.

5.1 Formulation

CNTs flow via nonlinear stretched sheet of deformable thickness with stagnation point

is addressed here. CNTs liquid saturates the permeable medium. Diffusion species and

melting heat are incorporated. The sheet consistence is y = B (x+ b1)0.5(1−m). It is taken

that T∞ > Tm∗ . Water-CNTs and kerosene oil-CNTs are treated the nanoliquid. The

coordinates selected are presented in Fig. 5.1. The equations following boundary layer

65

Fig. 5.1: Geometry of problem.

approximation take place

∂u

∂x+∂v

∂y= 0, (5.1)

u∂u

∂x+v

∂u

∂y= Ue

dUe

dx+ νnf

(∂2u

∂y2

)− νnf

k∗u, (5.2)

u∂T

∂x+v

∂T

∂y= αnf

∂2T

∂y2, (5.3)

u∂a

∂x+v

∂a

∂y= DA

(∂2a

∂y2

)− krab

2,

u∂b

∂x+v

∂b

∂y= DB

(∂2b

∂y2

)+ krab

2, (5.4)

u = Uw(x) = U0 (x+ b1)m , v = 0, T = Tm∗ ,

DA∂a

∂y= ksa, DB

∂b

∂y= −ksa, at y = B(x+ b1)

1−m2 ,

u→ Ue(x) = Ue(x+ b1)m T → T∞, a→ a0, b→ 0, asy → ∞. (5.5)

66

Melting heat equation is

knf

(∂T

∂y

)y=B(x+b)

1−m2

= ρnf [λ3 + Cs(Tm∗ − T0)] v(x, y = B (x+ b)1−m

2 ). (5.6)

Following Eq. 2.9 [96] and CNTs properties are presented in Table 2.1.

Using

η =y(m+ 1

2

U0 (x+ b)m−1

νf)0.5 ψ = (

2

m+ 1νfU0 (x+ b)m+1)0.5F (η) , Θ(η) =

T − Tm∗

T∞ − Tm∗,

u =U0 (x+ b)m F ′ (η) , v = −(m+ 1

2νfU0 (x+ b)m−1)0.5

[F (η) + ηF ′ (η)

m− 1

m+ 1

],

g =a

a0, h =

b

a0, (5.7)

equation (5.1) is clearly satisfied and Eqs. (5.2− 5.5) are reduced:

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

)F ′′′ + FF ′′ − 2m

m+ 1(F ′)

2+

2m

m+ 1A2

− 2k1m+ 1

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

)F ′ = 0, (5.8)

knf/kf(1− ϕ+ ϕ

(ρcp)CNT

(ρcp)f

)Θ′′ + PrFΘ′ = 0, (5.9)

1

Scg′′ + fg′ − 2K

m+ 1gh2 = 0, (5.10)

δ1Sch′′ + fh′ +

2K

m+ 1gh2 = 0. (5.11)

67

with

F ′(α) = 1, Θ(α) = 0,knf

kfM Θ′ (α) +

(1− ϕ+ ϕρCNT

ρf

)Pr[F (α) + m−1

m+1α] = 0,

ζ ′ (α) =√

2m+1

Ksg (α) , δ1h′ (α) = −

√2

m+1Ksg (α) , at α = B

√m+12

U0

νf,

F ′(∞) → A, Θ(∞) → 1, h (∞) → 0, g (∞) → 1 as α → ∞.

For same diffusion coefficients DA and DB, one has

g (α) + h (α) = 1.

Eqs. (5.11) and (5.12) yield

g′′ + Scfg′ − 2

m+ 1KScg(1− g)2 = 0. (5.12)

The parameters are defined as follows:

M =Cf (T∞ − Tm∗)

λ3 + Cs (Tm∗ − T0), P r =

µf (cp)fkf

, A =Ue

U0

, k1 =vf

kU0 (b1 + x)m−1 ,

K =kra

20 (b+ x)

Uw

, Ks =ksDA

√(b1 + x) νf

Uw

, Sc =νfDA

, δ1 =DB

DA

. (5.13)

68

Here α = B√

m+12

U0

νfis the wall thickness parameter. Defining Θ(η) = θ(η − α) = θ(η),

F (η) = f(η − α) = f(η), g(η) = ζ(η − α) = ζ(η), Eqs.(5.9− 5.14) become

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

) f ′′′ + ff ′′ − 2m

m+ 1(f ′)

2+

2m

m+ 1A2

− 2k1m+ 1

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

) f ′ = 0, (5.14)

knf/kf(1− ϕ+ ϕ

(p)CNT

(ρcp)f

) θ′′ + Pr fθ′ = 0, (5.15)

ζ ′′ + Scfζ ′ − 2

m+ 1ScKζ(1− ζ)2 = 0, (5.16)

f ′(0) = 1, ζ′(0) =

√2

m+ 1Ksζ(0), θ (0) = 0,

Mknfkf

θ′ (0) +

(1− ϕ+ ϕ

ρCNT

ρf

)(Pr f (0) +

m− 1

m+ 1α) = 0,

f ′(∞) → A, θ (∞) → 1, ζ (∞) → 1, asη → ∞. (5.17)

Skin friction coefficient and local Nusselt number are

Cf =τwρfU2

w

, Nux =(x+ b1) qw

kf (T∞ − Tm∗), (5.18)

τw = µnf

(∂u

∂y

)y=B(x+b1)

1−m2

, qw = −knf(∂T

∂y

)y=B(x+b1)

1−m2

. (5.19)

The above two expressions yield

CfRe1/2x =

1

(1− ϕ)12

√m+ 1

2f ′′(0), NuxRe

−1/2x = −knf

kf

√m+ 1

2θ′(0), (5.20)

where Rex = Uw (x+ b) νf .

69

5.2 OHAM outcomes

The functions (i.e. f0(η), θ0(η), ζ0(η)) and operators are defined by

f0(η) = Aη + (1− A)(1− exp(−η))−knf

kfM

(1− ϕ+ ϕρCNT

ρf)Pr

− αm− 1

m+ 1,

θ0(η) = 1− exp(−η), ζ0(η) = 1− 1

2exp(−

√2

m+ 1Ksη), (5.21)

Lf (f) =d3f

dζ3− df

dζ, Lθ(θ) =

d2θ

dη2− θ, Lζ(ζ) =

d2ζ

dη2− ζ, (5.22)

Lf [A1 + A2 exp(η) + A3 exp(−η)] = 0, (5.23)

Lθ [A4 exp(η) + A5 exp(−η)] = 0, (5.24)

Lζ [A6 exp(η) + A7 exp(−η)] = 0, (5.25)

where Ai the arbitrary constants. For optimal results of variables hf , hθ and hζ , we defined

the average squared residual errors at kth order

εfk (hf ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη

]2, (5.26)

εθk (hf , hθ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη ,k∑

i=0

(θi)η=jπη

]2, (5.27)

εζk (hf , hζ) =1

N + 1

N∑J=0

[k∑

i=0

(fi)η=jπη ,k∑

i=0

(ζi)η=jπη

]2, (5.28)

εt = εfk + εθk + εζk. (5.29)

Where εt denotes the total square residual error. The obtained results of optimal variables

via SWCNTs liquid as hf = −0.58073, hθ = −0.10347 and hζ = −1.02469 while optimal

results via MWCNTs liquid are hf = −0.507952, hθ = −0.100342, hζ = −1.99947. The

optimal results via kerosene SWCNTs liquid are hf = −0.158693, hθ = −0.020121 and

70

SWCNT-Water

4 6 8 10 12 14k

2´ 10-6

5´ 10-6

1´ 10-5

2´ 10-5

5´ 10-5

1´ 10-4error

(a)

SWCNT-Kerosene oil

4 6 8 10 12k

2´ 10-6

5´ 10-6

1´ 10-5

2´ 10-5

5´ 10-5

1´ 10-4error

(b)

Fig. 5.2: Total errors via SWCNTs liquid (a) and kerosene SWCNTs liquid (b).

MWCNT-Water

10 12 14 16 18k

0.0000100

5.0000000´ 10-6

2.0000000´ 10-6

3.0000000´ 10-6

1.5000000´ 10-6

7.0000000´ 10-6

error

(a)

MWCNT-Kerosene oil

4 6 8 10 12k

2´ 10-6

5´ 10-6

1´ 10-5

2´ 10-5

5´ 10-5

1´ 10-4error

(b)

Fig. 5.3: Total errors via MWCNTs water liquid (a) and kerosene MWCNTs liquid(b).

hζ = −0.847639 and hf = −0.216209, hθ = −0.031078 and hζ = −0.873847 are

the outcomes of kerosene MWCNTs liquid. Figs.(5.2-5.3) address the residual graphical

outcomes.

71

Table 5.1: Average square residual errors via M = α = A = ϕ = k1 = 0.1, m = 2,K = 0.4, Ks = 0.9 and Sc = 1.2.

SWCNT MWCNTk εfk εθk εζk εfk εθk εζk

4 (Water) 1.06×10−5 2.07×10−2 9.33×10−5 1.55×10−6 2.23×10−2 1.17×10−6

(Kerosene oil) 2.65×10−2 1.24 4.70×10−4 7.11×10−3 1.29×10−1 3.86×10−4

6 (Water) 3.94×10−8 1.64×10−3 2.10×10−5 3.49×10−8 1.21×10−3 1.16×10−6

(Kerosene oil) 2.16×10−3 5.40×10−2 2.78×10−4 3.28×10−4 3.42×10−2 2×10−4

10 (Water) 1.44×10−11 1.29×10−5 2.90×10−6 2.75×10−10 1.11×10−3 1.07×10−6

(Kerosene oil) 6.57×10−5 1.51×10−2 1.64×10−4 3×10−6 5.87×10−3 9.54×10−5

14 (Water) 2.61×10−13 1.34×10−6 1.29×10−6 1.23×10−12 1.01×10−3 1.03×10−6

(Kerosene oil) 2.66×10−6 3.57×10−3 1.55×10−4 1.36×10−8 1.26×10−3 9.41×10−5

16 (Water) 2.66×10−15 1.02×10−7 1.11×10−6 1.44×10−14 5.05×10−4 1×10−6

(Kerosene oil) 8.47×10−8 3.63×10−4 1.43×10−4 1.23×10−9 3.06×10−4 9.08×10−5

Table 5.2: Validation of f ′′ (0) with [100], [101] and [102] via A when k1 = ϕ = Ks =K = Sc = 0.

A [100] [101] [102] PR0.1 -0.9694 -0.9694 -0.969386 -0.9693790.2 -0.9181 -0.9181 -0.9181069 -0.91810580.5 -0.6673 -0.6673 -0.667263 -0.667262

Table 5.3: Validation of f ′′(0) via k1 when A = 0, ϕ = 0 [97].

k1 [97] PR0.5 1.22474487 1.22471 1.41421356 1.41421.5 1.58113883 1.58112 1.73205081 1.73205 2.44948974 2.4494

72

5.3 Discussion

The influence of variables on the outlines ( i.e. f ′ (η), θ (η), ζ (η)) via CNts-water and

CNTs-kerosene oil nanoliquid are explored here. The results of dimensionless variables i.e.

ϕ = M = A = k1 = 0.1, K = 0.4, Ks = 1.2 and Sc = 1.5. These parameters are fixed

except the variable in figures. Pr = 6.2 is fixed for water and kerosene oil as Pr = 21

throughout the study. Figs. 5.4 (a-b) address the variations of α on the velocity outline

via the CNTs liquid flow. Figs. 5.4(a-b) are sketched via water-CNTs and kerosene oil-

CNTs nanoliquids. Larger values of α enhances the velocity outline. Larger α via m < 1

correspond to enhance the thickness and ultimately velocity outline declines. Larger α via

m > 1 results enhances the velocity outline (see Figs. 5.5(a-b)). For m > 1 thickness

of wall declines via larger α. Figs. 5.6 (a-b) present the velocity outline via larger k1.

Larger k1 decline the velocity outline of CNTs liquid. The velocity outline increases via

higher ϕ (see Figs. 5.7(a-b)). Figs. 8.8 (a-b) address the curves of velocity outline via A.

The velocity outline enhances via A less than one and decline via A greater than one. For

A = 1, there is no velocity outline. The velocity outline dominates via water-CNTs liquid

than kerosene oil-CNTs liquid.

LargerM decline the temperature outline (see Figs. 5.9(a-b)). Fig. 5.10 (a-b) represent

that temperature of CNTs liquid increases via ϕ. The temperature outline enhances via

larger A (see Figs. 5.11(a-b)). The temperature outline develops high via MWCNTs-

liquid than SWCNTs-liquid. Larger A means higher free stream velocity which tends

to enhances the dynamic force and the temperature outline increases. The concentration

outline declines via K (see Figs. 5.12(a-b)). Figs 5.13− 5.14(a-b) address that larger Ks

and Sc tend to enhance the concentration outline.

73

Skin friction via larger A and ϕ is addressed in Fig. 5.15(a). Skin friction increases

via ϕ and opposite trend is noted via A. Skin friction via MWCNTs liquids is larger than

SWCNTs liquids. Fig. 5.15(b) reflects Nusselts number outcome via ϕ and M . Larger

ϕ enhances the heat transfer rate while it declines via M . Nusselt number in SWCNTs

liquid is noted higher than MWCNTs liquid. Table 5.1 presents residual errors via. Larger

order the error reduces. Table 5.2 addresses the validation of f ′′ (0) with [99-102]. The

outcomes match in favorable. Table 5.3 depicts f ′′(0) validation via published data.

74

η

f’(η

)

0 2 4 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

(a)η

f’(η

)

0 2 4 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α = 0.0, 0.5, 1.0

SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.4: Plots via α with m = 0.5 for f ′ (η).

η

f’(η

)

0 1 2 3 4 5 6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

(a)η

f’(η

)

0 1 2 3 4 5 6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α = 0.0, 0.5, 1.0

SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.5: Plots via α with m = 5 for f ′ (η).

75

η

f’(η

)

0 1 2 3 4 5 6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k1 = 0.0, 0.5, 1.0

SWCNT-Water

MWCNT-Water

(a)η

f’(η

)

0 1 2 3 4 5 6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SWCNT-Kerosene oil

MWCNT-Kerosene oil

k1 = 0.0, 0.5, 1.0

(b)

Fig. 5.6: Plots via k1 for f ′ (η).

η

f’(η

)

0 1 2 3 4 5 6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ= 0.0, 0.03, 0.1

SWCNT-Water

MWCNT-Water

(a)η

f’(η

)

0 1 2 3 4 5 6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ= 0.0, 0.03, 0.1

SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.7: Plots via ϕ for f ′ (η).

76

η

f’(η

)

0 1 2 30.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

A = 0.8

A = 1.0

A = 1.2SWCNT-Water

MWCNT-Water

(a)η

f’(η

)

0 1 2 30.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

A = 0.8

A = 1.0

A = 1.2SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.8: Plots via A for f ′ (η).

η

θ(η

)

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 0.0, 0.5, 0.7

SWCNT-Water

MWCNT-Water

(a)η

θ(η

)

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 0.0, 0.5, 0.7

SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.9: Plots via M for θ (η).

77

η

θ(η

)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ= 0.0, 0.1, 0.2

SWCNT-Water

MWCNT-Water

(a)η

θ(η

)

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ= 0.0, 0.1, 0.2

SWCNT-Kerosene oilMWCNT-Kerosene oil

(b)

Fig. 5.10: Plots via ϕ for θ (η).

η

θ(η

)

0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A = 0.8, 1.0, 1.2

SWCNT-Water

MWCNT-Water

(a)η

θ(η

)

0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

A = 0.8, 1.0, 1.2

SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.11: Plots via A for θ (η).

78

η

ζ(η

)

0 2 4 6 80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K = 0.4, 0.7, 1.0

SWCNT-Water

MWCNT-Water

(a)η

ζ(η

)

0 2 4 6 80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K = 0.4, 0.7, 1.0

SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.12: Plots via K for ζ (η).

η

ζ(η

)

0 2 4 6 8

0.4

0.5

0.6

0.7

0.8

0.9

1

Ks = 0.9, 1.2, 1.5

SWCNT-Water

MWCNT-Water

(a)η

ζ(η

)

0 2 4 6 8

0.4

0.5

0.6

0.7

0.8

0.9

1

Ks = 0.9, 1.2, 1.5

SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.13: Plots via Ks for ζ (η).

79

η

ζ(η

)

0 1 2 3 4 5 6 7 8

0.4

0.5

0.6

0.7

0.8

0.9

1

Sc = 0.7, 1.2, 1.7

SWCNT-Water

MWCNT-Water

(a)η

ζ(η

)

0 1 2 3 4 5 6 7 8

0.4

0.5

0.6

0.7

0.8

0.9

1

Sc = 0.7, 1.2, 1.7

SWCNT-Kerosene oil

MWCNT-Kerosene oil

(b)

Fig. 5.14: Plots via Sc for ζ (η).

A

CfR

ex1

/2

0 0.1 0.2 0.3

-4

-3.5

-3

-2.5

-2

SWCNT-WaterMWCNT-WaterSWCNT-Kerosene oilMWCNT-Kerosene oil

φ= 0.3, 0.4

(a)M

Nu

xR

ex-1

/2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

SWCNT-WaterMWCNT-WaterSWCNT-Kerosene oilMWCNT-Kerosene oil

φ= 0.4, 0.5

(b)

Fig. 5.15: (a) Plots for skin friction, (b) Plots for Nusselt number.

80

5.4 Main findings

CNTs Flow of nanoliquids via nonlinear stretched surface of deformable thickness has

been investigated. Major key findings are as:

• The CNTs flow enhances via larger α (i.e. m > 1) and ϕ and it reduces for via α

(i.e. m < 1) and k1 for water-CNTs liquid and kerosene oil-CNTs liquid. Kerosene

oil nanoliquid dominates water nanoliquid.

• Temperature outline declines via larger M . Influence of M in SWCNTs liquids is

larger than MWCNTs liquids.

• Variables K and Ks reflects reverse trend in water-CNTs liquid and kerosene oil-

CNTs liquid.

• Larger A declines the skin friction and it enhances via larger ϕ.

• Larger ϕ tends to higher rate of heat and inverse behavior is seen via larger M .

81

Chapter 6

Convective flow of Carbon nanotubes via three-dimensional

Three-dimensional flow of canbon nanotubes is investigated with porous medium.

Convective conditions and non-uniform heat generation is incorporated for heat transfer.

Diffusion species is accounted for concentration. Numerical outcomes are developed via

shooting technique (i.e. Runge-Kutta method of fifth order). Heat transfer is dominated in

SWCNTs fluid when compared with MWCNTs fluid.

6.1 Formulation

Consider carbon nanotubes flow via permeable medium. 3D(three-dimensional) flow

via nonlinear stretched sheet. uw = a1(x + y)n (in x − direction), vw = b1(x + y)n (in

y − direction) present nonlinear sheet velocities with constants a1, b1, n ≻ 0 (see Fig.

6.1). Temperature at surface is balanced via hotted liquid with temperature Tf through

convection. The homogenous equation is represented by

A+ 2B → 3B, rate = krab2. (6.1)

82

Fig. 6.1: Physical coordinates.

Heterogeneous equation is

A → B, rate = ksa. (6.2)

The relevant problems are

∂w

∂z+∂v

∂y+∂u

∂x= 0, (6.3)

w∂u

∂z+v

∂u

∂y+ u

∂u

∂x= νnf

(∂2u

∂z2

)− νnf

k∗u, (6.4)

w∂v

∂z+v

∂v

∂y+ u

∂v

∂x= νnf

(∂2v

∂z2

)− νnf

k∗v, (6.5)

w∂T

∂z+v

∂T

∂y+ u

∂T

∂x= αnf

∂2T

∂z2+Q(z)

ρCp

(T − T∞), (6.6)

w∂a

∂z+v

∂a

∂y+ u

∂a

∂x= DA

(∂2a

∂z2

)− krab

2, (6.7)

w∂a

∂z+v

∂b

∂y+ u

∂b

∂x= DB

(∂2b

∂z2

)+ krab

2. (6.8)

83

with the constrains

u = uw = a1 (x+ y)n , v = vw = b1 (x+ y)n , w = 0,

− knf∂T

∂z= hf (Tf − T ), DA

∂a

∂z= ksa, DB

∂b

∂z= −ksa at z = 0,

u→ 0, v → 0 T → T∞ a→ a0, b→ 0 as z → ∞. (6.9)

In above expressions Q(z) = (x + y)n−1Q0 is non-uniform heat source. Following Xue

(Eq. 6.10) and renovated H-C (Eq. 6.11) models, one has

µnf =µf

(1− ϕ)2.5, νnf =

µnf

ρnf, ρnf = (1− ϕ) ρf + ϕρCNT ,

αnf =knf

ρnf (cp)nf,

knfkf

=(1− ϕ) + 2ϕ kCNT

kCNT−kfln

kCNT+kf2kf

(1− ϕ) + 2ϕ kfkCNT−kf

lnkCNT+kf

2kf

, (6.10)

knfkf

=kpe + kf (n− 1) + (n+ 1)(kpe − kf )(1 + C)ϕ

kpe + kf (n− 1)− (kpe − kf )(1 + C)ϕ,

kpe =1

π

∫ π

0

√k2pez sin

2 θ + k2pex cos2 θdθ, kpex =

A1ϕkCNT +B1Cϕklr

A1ϕ+B1Cϕ,

kpez =kCNT + Cklr

1 + C, klr =

kCNTR1(1 + t1/R1 − kf/kCNT ) ln(1 + t1/R1)

tkf ln[(1 + t1/R1)kCNT/kf ],

A1 = − 2klr

kCNTklr, B1 =

R1

R1 + t1

(kCNT − klr

klr + kCNT

− 1

), C =

(R1 + t1)2 −R2

1

R21

,

n1 = 3ψ−g2 , ψ =2e(u)[1− e(u)2]1/6

e(u)(1− e(u)2 arcsin(e(u)))0.5, e(u) = (1− R2

1 + u

(L1/2)2 + u)0.5.

(6.11)

Assuming

η =a1(n+ 1)

2νf

0.5

(x+ y)0.5(n−1) z, u = a1(x+ y)nf ′(η), v = b1(x+ y)ng′(η),

w = −a1νf (n+ 1)

2

0.5

(x+ y)0.5(n−1)

((f + g) +

n− 1

n+ 1η (f ′ + g′)

),

θ =T − T∞Tf − T∞

, ζ =a

a0, h =

b

a0. (6.12)

84

Eqs. (6.4− 6.11) become

(1

(1− ϕ)5/2(1− ϕ+ ϕρCNT

ρf))f ′′′ + (f + g)f ′′ − 2n

n+ 1(f ′ + g′)f ′

− 2k1n+ 1

(1

(1− ϕ)5/2(1− ϕ+ ϕρCNT

ρf))f ′ = 0, (6.13)

(1

(1−ϕ)5/2(1−ϕ+ϕρCNT

ρf))g′′′ + (f + g)g′′ − 2n

n+ 1(f ′ + g′)g′

− 2k1n+ 1

(1

(1−ϕ)5/2(1−ϕ+ϕρCNT

ρf))g′ = 0, (6.14)

(knf/kf

(1− ϕ+ ϕ (ρcp)CNT

(ρcp)f))θ′′ + Pr((f + g)θ′ +

2δ

n+ 1θ) = 0, (6.15)

1

Scζ ′′ + (f + g)ζ ′ − 2K

n+ 1ζh2 = 0, (6.16)

δ1Sch′′ + (f + g)h′ +

2K

n+ 1ζh2 = 0, (6.17)

f ′(0) =1, g′(0) = α1, f(0) = 0, g(0) = 0, θ′(0) = − kfknf

Bi (1− θ(0)) ,

ζ ′ (0) =(2

n+ 1)0.5Ksζ (0) , δ1h

′ (0) = −(2

n+ 1)0.5Ksζ (0) ,

f ′(∞) =g′(∞) → 0, ζ (∞) → 1, h (∞) → 0, ζ (∞) → 0. (6.18)

Choosing DA = DB i.e. δ1 = 1.

ζ (η) + h (η) = 1, (6.19)

Eqs. (6.16− 6.17) are simplified to

ζ ′′ + Sc(f + g)ζ ′ − 2

n+ 1KScζ(1− ζ)2 = 0, (6.20)

ζ ′(0) =2

n+ 1

0.5

Ksζ(0), ζ(∞) → 1. (6.21)

85

The definitions of variables are

k1 =νf (x+ y)

k∗uw, δ =

Q0

ρa1Cp

, P r =νfαf

, Bi =hfkf

√n+ 1

2

(x+ y)νfuw

, α1 =b1a1,

K =kra

20 (x+ y)

uw, Ks =

ksDA

√(x+ y) νf

uw, Sc =

νfDA

, δ1 =DB

DA

. (6.22)

Skin friction and local Nusselt number are

Cfx =τzxρfu2w

, Cfy =τzyρfv2w

, Nux =(x+ y)qwk(Tw − T∞)

, (6.23)

where

τzx = µ

(∂u

∂z+∂w

∂x

)z=0

, τzy = µ

(∂v

∂z+∂w

∂y

)z=0

, qw = −k(∂T

∂z

)z=0

. (6.24)

Now

Re12xCfx =

(n+ 1

2

)0.51

(1− ϕ)5/2f ′′(0), Re

12yCfy = α− 3

2

(n+ 1

2

)0.51

(1− ϕ)5/2g′′(0),

Re−12

x Nux = −knfkf

√n+ 1

2θ′(0), (6.25)

where Rex = uw(x+y)νf

and Rey =vw(x+y)

νf.

6.2 Shooting technique results

Shooting result via Runge-Kutta of order 5th integration is developed via outcomes of

Eqs(6.13 − 6.21). Computation is carried via Matlab software. Shooting method deals

86

only initial values equations. Therefore Eqs(6.13− 6.21) become

f ′′′ =− ((1− ϕ)5/2(1− ϕ+ ϕρCNT

ρf))(f + g)f ′′ − 2n

n+ 1(f ′ + g′)f ′

− 2k1n+ 1

(1

(1−ϕ)5/2(1−ϕ+ϕρCNT

ρf))f ′, (6.26)

g′′′ =− ((1− ϕ)5/2(1− ϕ+ ϕρCNT

ρf))((f + g)g′′ − 2n

n+ 1(f ′ + g′)g′

− 2k1n+ 1

(1

(1−ϕ)5/2(1−ϕ+ϕρCNT

ρf))g′), (6.27)

θ′′ =− ((1− ϕ+ ϕ (ρcp)CNT

(ρcp)f)

knf/kf)

[Pr((f + g)θ′ +

2δ

n+ 1θ)

], (6.28)

ζ ′′ =− Sc(f + g)ζ ′ − 2

n+ 1KScζ(1− ζ)2. (6.29)

87

Thus Eqs(6.26− 6.29) are converted into first order ODE’s . The equations reduced to

y′1 =y2, (6.30)

y′2 =y3, (6.31)

y′3 =− ((1− ϕ)5/2(1− ϕ+ ϕρCNT

ρf))((y1 + y4)y3 −

2n

n+ 1(y2 + y5)y2

− 2k1n+ 1

(1

(1−ϕ)5/2(1−ϕ+ϕρCNT

ρf))y2), (6.32)

y′4 =y5, (6.33)

y′5 =y6, (6.34)

y′6 =− ((1− ϕ)5/2(1− ϕ+ ϕρCNT

ρf))((y1 + y4)y6− 2n

n+ 1(y2 + y5)y5

− 2k1n+ 1

(1

(1−ϕ)5/2(1−ϕ+ϕρCNT

ρf))y5), (6.35)

y′7 =y8, (6.36)

y′8 =− ((1− ϕ+ ϕ (ρcp)CNT

(ρcp)f)

knf/kf)

[Pr

((y1 + y4)y8 +

2δ

n+ 1y7)

], (6.37)

y′9 =y10, (6.38)

y′10 =−[Sc(y1 + y4)y10 −

2

n+ 1KScy9(1− y9)

2

], (6.39)

with the associated constrains

y1(0) = y4(0) = 0 , y2(0) = 1, y5(0) = α, y2(∞) → 0, y5(∞) → 0,

y8(0) = − kfknf

γ(1− y7(0)), y10(0) =

√2

n+ 1Ksy9(0), y7(∞) → 0 y9(∞) → 1.

(6.40)

Shooting technique is implemented for the solution. In this technique choose the limit for

η∞ first and this value is i.e η∞ is 7. Find initial guesses for y3(0), y6(0), y8(0) and y10(0).

88

Initially set y3(0) = y6(0) = y8(0) = y10(0) = −1 in second step and the Runge-Kutta

fifth order is used to solve the system of first order ODE’s thirdly. Boundary residuals is

evaluated in last step. The results converge in case absolute results of boundary residuals

are below then 10−6 (i.e. tolerance error ). If residuals are found greater than 10−6 in that

case results of y3(0), y6(0), y8(0) and y10(0) are revise via Newtons method. The processes

is carried it results are below 10−6. Its benefit over other method is fifth order truncation

error.

6.3 Discussion

Physical explanation of variables used in problem is disclosed here. These variables

are kept constant (i.e. ϕ = Bi = α1 = k1 = δ = 0.1, n = 2, Pr = 6.2, K = 0.4,

Ks = 1 and Sc = 1.2) except the variable mention in figures. CNTs liquid properties

are addressed in Table 2.1. The numerical outcomes of skin friction is depicted for ϕ and

k1 in Table 6.1. The shear stresses at wall enhance via k1 and ϕ. Heat transfer controls

via larger δ and enhances via ϕ and Bi. SWCNTs liquid shows higher heat transfer than

MWCNTs liquid. Table 6.2 reveals results via Xue and renovated H-C models. Thermal

conductivity via renovated H-C model noted higher than Xue model. In renovated H-C

model interfacial layer, aspect ratio and CNT’s diameter are accounted. Results of f ′′(0)

via k1 is given in Table 6.3 for comparison. This table also reflects outcomes of shooting

technique and Bvp4c. Results are in favorable.

Fig. 6.2 is sketched via ϕ for velocity. The flow of SWCNTs liquid and MWCNTs

liquid boost in x− and y− directions. The velocity outline declines via k1 (see Fig. 6.3) in

both axes. Curves via n ( power index) is shown in Fig. 6.4. The liquid flow enhances via

larger n. Larger α1 declines stretching rate in x− and enhances in y− directions (see Fig.

89

6.5). Hence the liquid flow declines via x− axis and enhances via y− axis. The velocity

for SWCNTs liquid in observed higher than MWCNTs liquid.

The temperature outline via δ is addressed in Fig. 6.6. Larger δ corresponds higher

temperature. It is seen that temperature outlines increase for CNTs liquid via higher Bi

(see Fig. 6.7). Fig. 6.8 addressed temperature curves via ϕ. Thermal layer thickness

enhances via larger phi. Fig. 6.9 reflects that larger α1 declines the temperature outlines.

Temperature in SWCNTs liquid is noted higher than MWCNTs liquid.

Concentration outline via K is sketched in Fig. 6.10 for CNTs liquid. Larger K

declines the concentration. The concentration improves via lager Ks and Sc (see Figs.

6.11 − 6.12). Fig. 6.13 represents the comparison via Xue and renovated H-C models for

temperature curves. The temperature curves are higher via renovated H-C model. Rate

of heat transfer for Xue and renovated H-C models is plotted via Fig. 6.14. Higher val-

ues of Bi and ϕ lead to higher heat transfer magnitude. Heat transfer via renovated H-C

model is improved than via Xue model. Further heat transfer in SWCNTs liquid is higher

than MWCNTs liquid. In 2D case the streamlines for CNTs liquid are presented in figs

6.15− 6.16 while isotherms in figs 6.17− 6.18.

90

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f′ (η),

g′ (η

)

SWCNT

MWCNT

g′(η)

f′(η)

φ = 0.1, 0.2, 0.3

Fig. 6.2: Plots via ϕ for f ′(η).

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f′ (η),

g

′ (η)

k1 = 0.1, 0.5, 0.9g′(η)

f′(η)MWCNT

SWCNT

Fig. 6.3: Plots via k1 for f ′(η).

91

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f′ (η),

g

′ (η)

n = 1, 4g′(η)

f′(η)

MWCNT

SWCNT

Fig. 6.4: Plots via n for f ′(η).

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f′ (η

), g

′ (η

)

f′(η)

g′(η)

SWCNT

MWCNT

α1 = 0.1, 0.3, 0.5

Fig. 6.5: Plots via α1 for f ′(η).

92

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

2.6 2.8 3 3.2 3.4

0.15

0.2

0.25

0.3

0.35

0.4

ηθ

( η

)

SWCNT

MWCNT

δ = 0.1, 0.2, 0.3

Fig. 6.6: Plots via δ for θ(η).

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

2.4 2.6 2.8 3 3.2

0.1

0.15

0.2

0.25

0.3

0.35

η

θ (

η )

SWCNT

MWCNT

Bi = 0.1, 0.3, 0.7

Fig. 6.7: Plots via Bi for θ(η).

93

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η)

φ = 0.1, 0.2, 0.3

SWCNT

MWCNT

Fig. 6.8: Plots via ϕ for θ(η).

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

θ (

η )

3 3.2 3.4 3.6 3.8

0.05

0.1

0.15

0.2

0.25

η

θ(η

)

SWCNT

MWCNT

α1 = 0.1, 0.2, 0.3

Fig. 6.9: Plots via α1 for θ(η).

94

0 1 2 3 4 5 6

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ζ (

η)

K = 0.4, 0.8, 1.2

SWCNT

MWCNT

Fig. 6.10: Plots via K for ζ(η).

0 1 2 3 4 5 60.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ζ (

η)

SWCNT

MWCNT

Ks = 0.5, 0.9, 1.5

Fig. 6.11: Plots via Ks for ζ(η).

95

0 1 2 3 4 5 60.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ζ(η

)

SWCNT

MWCNT

Sc = 0.7, 1.2, 1.5

Fig. 6.12: Plots via Sc for ζ(η).

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

θ (

η

)

1.8 2 2.2 2.4 2.60

0.05

0.1

0.15

0.2

0.25

η

θ(η

)

Xue modelSWCNTMWCNT

renovatedHamilton−Crosser modelSWCNTMWCNT

k1 = 0.1

n = 2,δ = 0.1,φ = 0.1,α

1 = 0.1,

Bi = 0.1,K = 0.4,Ks = 1,Sc = 1.2

Fig. 6.13: Outlines for Xue and renovated H-C models.

96

γ

Re x-0

.5N

u x

0.2 0.4 0.6 0.80

1

2

3

4

5

6

φ= 0.01

φ= 0.02

φ= 0.03

φ= 0.01

φ= 0.02

φ= 0.03

renovated

Hamilton-Crosser model

Xue model

Fig. 6.14: Plots via Bi and ϕ for Xue and renovated H-C models.

η

-4 -2 0 20

1

2

3

4

5

6

7

V

543210

-1-2-3-4-5

Fig. 6.15: Plots of streamlines for SWCNTs.

97

η

V

2.4392

1.12765

0.56105

-0.787536

-5.41107

Fig. 6.16: Plots of streamlines for MWCNTs.

z

η

1 1.5 2 2.5 30

0.5

1

1.5

2

V: 0 0.000216283 0.02 0.155755

Fig. 6.17: Plots of isotherms for SWCNTs.

98

z

η

V: 8.32538E-05 0.00167042 0.0147382

Fig. 6.18: Plots of isotherms for MWCNTs.

99

Table 6.1: Numerical values of skin friction Re12xCfx and Re

12yCfy for SWCNTs liquid and

MWCNTs liquid.

SWCNT MWCNT

ϕ k1 −Re12xCfx −Re

12xCfy −Re

12xCfx −Re

12xCfy

0.1 0.1 0.2446 0.0045 0.0352 0.07650.2 0.3121 0.0132 0.1045 0.08500.3 0.3412 0.0152 0.1640 0.09100.1 0.3 0.5630 0.0241 0.4452 0.0249

0.5 0.7453 0.0393 0.6962 0.04950.7 0.8030 0.0518 0.8673 0.0696

Table 6.2: Resutls for Re−12Nux via ϕ, δ and γ in case of CNTs liquid.

Xue [96] Yang and Xu [98]ϕ δ γ Re−

12Nux (SWCNT) Re−

12Nux (MWCNT) Re−

12Nux (SWCNT) Re−

12Nux (MWCNT)

0.1 0.1 0.1 1.0334 0.9311 7.3147 7.25620.3 4.3048 3.9074 14.8892 14.66710.5 9.4089 8.5755 35.6698 34.39220.1 0.1 0.1 1.0334 0.9311 7.3147 7.2562

0.3 0.9853 0.8915 4.1760 4.19400.4 0.9019 0.8232 0.1644 0.2875

0.1 0.1 0.1 1.0334 0.9311 7.3147 7.25620.3 2.1585 1.9904 8.2751 8.23310.5 2.7593 2.5768 8.4983 8.4609

Table 6.3: Validation of f ′′(0) via k1 and ϕ = 0, [97], [104].

k1 [97] [104] PR0.5 1.22474487 1.2247 1.2247421 1.41421356 1.4142 1.414211

1.5 1.58113883 1.5811 1.5811372 1.73205081 1.7320 1.7320595 2.44948974 2.4494 2.449486

100

6.4 Main findings

Convective carbon nanotubes flow is investigated. Major findings are

• CNTs liquid flow enhances in y− line and it declines in x− line via 1. Larger n and

ϕ correspond to higher the velocity outlines while opposite trend is noticed via k1.

The velocity curves improve in MWCNTs liquid than that of SWCNTs liquid.

• Larger Bi and δ improve temperature outlines and declines via 1.

• Skin friction enhances via ϕ and k1. SWCNTs liquid offers more resistance than

MWCNTs liquid.

• Higher values of Bi and ϕ correspond less heat transfer and it improves by δ. Heat

transfer in SWCNTs liquid is noted higher than MWCNTs liquid.

101

Chapter 7

CNTs flow with slip condition

3D flow of carbon nanotubes via slendering non-linear stretched sheet is incorporated

in this chapter. CNTs liquid saturates the porous medium with slip impact. Diffusion

species are accounted for mass transfer. Outcomes are obtained via Bvp4c matlab solver.

Graphical findings and tabulated values are elaborated for the parameters in problem.

7.1 Formulation

Consider 3D (three-dimensional) carbon nanotubes flow with slip influence via de-

formable non-linear stretched sheet. z = B(x + y + c)n−12 the deformable sheet, uw =

a1(x+ y+ c)n and vw = b1(x+ y+ c)n the deformable sheet velocities in x− y directions

(see Fig. 7.1). Here B, a1, b1, c, n ≻ 0 are positive constants. The homogeneous equation

is

A+ 2B → 3B, rate = krab2. (7.1)

102

Fig. 7.1: Physical model

and the heterogeneous equation is

A → B, rate = ksa. (7.2)

The relevant problems take the forms of Eqs 6.3 − 6.8. Here the heat source is not

incorporated. The constrains are

u(x, y, z) =uw(x) + h∗1(∂u

∂z), v(x, y, z) = vw(x) + h∗1(

∂v

∂z),

T (x, y, z) =Tw(x) + h∗2(∂T

∂z), DA(

∂a

∂z) = ksa, DB(

∂b

∂z) = −ksa,

u→ 0, v → 0 T → T∞ a→ a0, b→ 0 as z → ∞. (7.3)

103

Here [105]

ξ1 =kBT√2πd21p

, h∗1 =

[2− f1f1

]ξ1(x+ y + c)

n−12 , ξ2 =

(2Γ

1 + Γ

)ξ1Pr

, (7.4)

h∗2 =

[2− b2b2

]ξ2(x+ y + c)

n−12 . (7.5)

Following Xue [96], Table [2.1]

µnf =µf

(1− ϕ)2.5, νnf =

µnf

ρnf, ρnf = (1− ϕ) ρf + ϕρCNT ,

αnf =knf

ρnf (cp)nf,

knfkf

=(1− ϕ) + 2ϕ kCNT

kCNT−kfln

kCNT+kf2kf

(1− ϕ) + 2ϕ kfkCNT−kf

lnkCNT+kf

2kf

, (7.6)

Letting [105];

η =

√a1(n+ 1)

2νf(x+ y + c)

n−12 z, u = a1(x+ y + c)nf ′(η), v = a1(x+ y + c)ng′(η),

w = −√

2a1νfn+ 1

(x+ y + c)n−12

[(n+ 1

2)(f + g) + (

n− 1

2)η (f ′ + g′)

],

θ =T − T∞Tw − T∞

, ζ =a

a0, h =

b

a0, (7.7)

104

equations (7.4− 7.6) become

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

) f ′′′ + (f + g) f ′′ − 2n

n+ 1(f ′ + g′) f ′

− 2k1n+ 1

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

) f ′ = 0, (7.8)

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

) g′′′ + (f + g) g′′ − 2n

n+ 1(f ′ + g′) g′

− 2k1n+ 1

1

(1−ϕ)5/2(1−ϕ+ϕ

ρCNTρf

) g′ = 0, (7.9)

knf/kf(1− ϕ+ ϕ

(ρcp)CNT

(ρcp)f

) θ′′ +

2

n+ 1Pr

(n+ 1

2(f + g) θ′

)= 0, (7.10)

1

Scζ ′′ + (f + g) ζ ′ − 2K

n+ 1ζh2 = 0, (7.11)

δ1Sch′′ + (f + g)h′ +

2K

n+ 1ζh2 = 0, (7.12)

f(0) =α

(1− n

1 + n

)[1 + h∗1f

′′(0)], f ′(0) = [1 + h∗1f′′(0)], (7.13)

g(0) =α

(1− n

1 + n

)[1 + h∗1g

′′(0)], g′(0) = [1 + h∗1g′′(0)], θ(0) = [1 + h∗2θ

′(0)],

(7.14)

ζ ′ (0) =(2

n+ 1)0.5Ksζ (0) , δ1h

′ (0) = −(2

n+ 1)0.5Ksζ (0) , (7.15)

f ′(∞) →0, g′(∞) → 0, ζ (∞) → 1, h (∞) → 0, ζ (∞) → 0. (7.16)

Selecting DA = DB (i.e. δ1 = 1) here

ζ (η) + h (η) = 1, (7.17)

105

Eqs. (7.16), (7.17) and (7.20) become

ζ ′′ + Sc(f + g)ζ ′ − 2

n+ 1KScζ(1− ζ)2 = 0, (7.18)

ζ ′(0) = (2

n+ 1)0.5Ksζ(0), ζ(∞) → 1, (7.19)

The definition of parameters in above equations are

k1 =νf (x+ y + c)

k∗uw, P r =

νfαf

, α = B

√(n+ 1)a1

2νf, h1 = h∗1

√(n+ 1)a1(x+ y + c)n−1

2νf,

h2 =h∗2

√(n+ 1)a1(x+ y + c)n−1

2νf, K =

kra20 (x+ y + c)

uw, Ks =

ksDA

√(x+ y + c) νf

uw,

Sc =νfDA

, δ1 =DB

DA

. (7.20)

Skin friction and local Nusselt number are

Re12xCf = 2

√n+ 1

2

1

(1− ϕ)52

f ′′(0), NuxRe− 1

2x = −knf

kf

√n+ 1

2θ′(0), (7.21)

in which Rex = uw(x+y+c)νf

.

7.2 Bvp4c outcomes

First the system of ODE’s convert into first order and then follow the steps by calling

the function sol = Bvp4c(odeEx,bcEx,solinit),

sol = Bvp4c(odeEx,bcEx,solinit),

solinit = bvpinit(eta, yinit, variable).

106

7.3 Discussion

Keep the values of variables fixed for numerical results as ϕ = 0.1, k1 = 0.1, Pr = 6.2,

α = 0.1, h1 = 0.1, n = 2, h2 = 0.1, K = 0.4, Ks = 1.0 and Sc = 1.2. The values of

these variables are assumed constant except the parameter given in figures. Outcomes via

CfRe1/2x and NuzRe

−1/2x are analyzed in Table 7.1. It is concluded that skin friction en-

hances via k1 and it can be overcome via h1, ϕ, α. Skin friction for SWCNTs liquid is

noted higher than MWCNTs liquid. Heat transfer rate becomes less via larger α. MWC-

NTs liquid has higher heat transfer rate than SWCNTs liquid. The validation of present

study is given with published data [97], [104] via k1 in Table 7.2.

Figs. 7.2 − 7.3 shows the plots via ϕ on velocity outline for CNTs fluid. ϕ directly

relates with CNTs liquid flow in both directions. k1 declines the CNTs liquid flow in both

directions (see Fig. 7.4 − 7.5). The velocity outline declines via h1 (see Fig. 7.6 − 7.7).

Fig. 7.8 − 7.9 addresses that larger α declines the CNTs liquid velocity. It is noticed that

MWCNTs liquid has higher velocity than SWCNTs liquid.

The temperature outline enhances via larger h1 (see Fig. 7.10). Fig. 7.11 depics that

larger h2 declines the temperature of liquid. Temperature for SWCNTs liquid is noted

higher than MWCNTs liquid. As the values of K is enhanced the concentration declines

(see Fig. 7.12). Figs. 7.13 − 7.14 represents that the concentration outline improves via

higher values ofKs and Sc. The concentration in MWCNTs liquid is noted slightly higher

than SWCNTs liquid.

107

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f ′ (η

)

φ = 0.1, 0.2, 0.3

MWCNT−Water

SWCNT−Water

Fig. 7.2: Plots via ϕ for f ′(η).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g ′ (

η)

φ = 0.1, 0.3, 0.5

SWCNT−Water

MWCNT−Water

Fig. 7.3: Plots via ϕ for g′(η).

108

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

f ′ (η

)

k1 = 0.1, 0.5, 0.9

SWCNT−Water

MWCNT−Water

Fig. 7.4: Plots via k1 for f ′(η).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

g ′ (

η)

k1 = 0.1, 0.5, 0.9

SWCNT−Water

MWCNT−Water

Fig. 7.5: Plots via k1 for g′(η).

109

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

f ′ (η

)

h1 = 0.1, 0.5, 0.9

SWCNT−Water

MWCNT−Water

Fig. 7.6: Plots via h1 for f ′(η).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

g ′ (

η)

h1 = 0.1, 0.5, 0.9

SWCNT−Water

MWCNT−Water

Fig. 7.7: Plots via h1 for g′(η).

110

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

f ′ (η

)

α = 0.1, 0.3, 0.7

SWCNT−Water

MWCNT−Water

Fig. 7.8: Plots via α for f ′(η).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

g ′ (

η)

α = 0.1, 0.3, 0.7

SWCNT−Water

MWCNT−Water

Fig. 7.9: Plots via α for g′(η).

111

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η)

h1 = 0.1, 0.5, 0.9

SWCNT−Water

MWCNT−Water

Fig. 7.10: Plots via h1 for θ(η).

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

θ (

η)

h2 = 0.1, 0.5, 0.9

SWCNT−Water

MWCNT−Water

Fig. 7.11: Plots via h2 for θ(η).

112

0 0.5 1 1.5 2 2.5 30.4

0.5

0.6

0.7

0.8

0.9

1

η

ζ (

η)

K = 0.1, 0.7, 1.5

SWCNT−Water

MWCNT−Water

Fig. 7.12: Plots via K for ζ(η).

0 0.5 1 1.5 2 2.5 3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ζ (

η)

Ks = 0.9, 1.2, 1.7

SWCNT−Water

MWCNT−Water

Fig. 7.13: Plots via Ks for ζ(η).

113

0 0.5 1 1.5 2 2.5 30.4

0.5

0.6

0.7

0.8

0.9

1

η

ζ (

η)

SWCNT−Water

MWCNT−Water

Sc = 0.8, 1.2, 1.5

Fig. 7.14: Plots via Sc for ζ(η).

Table 7.1: Skin friction and Nusselt number for CNTs liquid via ϕ, k1, α, h1 and h2.

SWCNT MWCNTϕ k1 h1 α h2 Re

12Cfx Re−

12Nux Re

12Cfx Re−

12Nux

0.1 0.1 0.1 0.1 0.1 1.29718 1.17954 1.25054 1.243820.2 1.21346 0.788291 1.3551 0.8517010.3 1.11098 0.598683 1.015 0.6485360.1 0.1 0.1 0.1 0.1 1.29718 1.17954 1.25054 1.24382

0.3 1.34576 1.16441 1.30153 1.228030.7 1.43632 1.1359 1.39619 1.19819

0.1 0.1 0.1 0.1 0.1 1.29718 1.17954 1.25054 1.243820.3 0.96110 1.04697 0.93509 1.108670.7 0.64915 0.88920 0.63686 0.94508

0.1 0.1 0.1 0.1 0.1 1.29718 1.17954 1.25054 1.243820.3 1.21346 1.55361 1.35511 1.640990.7 1.11098 2.25094 1.01503 2.37733

0.1 0.1 0.1 0.1 0.1 1.29718 1.17954 1.25054 1.243820.3 1.34576 0.95439 1.30153 0.996040.7 1.43632 0.69071 1.39619 0.71226

114

Table 7.2: Validation [97], [104] of f ′′(0) for k1 fixed h1 = 0, h2 = 0, ϕ = 0, α = 0,n = 1.

k1 [97] [104] PR0.5 1.22474487 1.2247 1.224741 1.41421356 1.4142 1.41421

1.5 1.58113883 1.5811 1.581132 1.73205081 1.7320 1.732055 2.44948974 2.4494 2.44948

7.4 Main findings

3D CNTs liquid flow with diffusion species via deformable sheet is addressed. The

findings are

• The velocity outlines improves via ϕ and reverse trend is noted via k1, h1, h2.

• CNTs liquid temperature improves via h1 and h2 declines the temperature outline.

K declines the concentration while Sc improves the concentration.

• Larger k1 enhances the skin friction and opposite trend is noted via α, ϕ and h1.

Magnitude of Heat transfer reduces via larger k1, ϕ, h1 and h2. Heat transfer rate

enhances for larger values of α. MWCNTs liquid has higher rate of heat transfer

when compared with SWCNTs liquid.

115

Chapter 8

Newtonian heating flow with diffusion species

Here Newtonian heating in flow of MHD Powell-Eyring liquid is discussed. Contribu-

tion for homogeneous-heterogeneous reactions are highlighted. Emphasis is given to the

momentum, energy and concentration. Outlines (i.e. f ′(η),θ(η),ζ(η)) are examined. Val-

idation with previous results is noticed. CfRe12z and NuzRe

−12

z are investigated. Clearly

flow accelerates for larger Powell-Eyring material parameter. Concentration for homoge-

neous and heterogeneous reactions is opposite.

8.1 Formulation

MHD Powell-Eyring liquid flow is considered via a stretched cylinder. Heat trans-

fer presents via heat generation/absorption, diffusion species and Newtonian heating. β0

(i.e. Uniform magnetic field) is accounted along radial axes (see Fig. 8.1). Low mag-

netic Reynolds number corresponds that impact of induced magnetic field is ignored. The

homogeneous reaction equation is

A+ 2B → 3B, rate = krab2, (8.1)

116

Fig. 8.1: Geometry of problem.

with isothermal reaction equation

A → B, rate = ksa. (8.2)

Applying boundary layer approximations, we have

∂ (rw)

∂z+∂ (ru)

∂r= 0, (8.3)

w∂w

∂z+ u

∂w

∂r= ν

(1

r

∂w

∂r+∂2w

∂r2

)+

1

ρβ3c1

(∂2w

∂r2+

1

r

∂w

∂r

)− 1

6ρβ3c31

(1

r

(∂w

∂r

)3

+ 3

(∂w

∂r

)2(∂2w

∂r2

))− σ1β

20

ρu, (8.4)

u∂T

∂r+ w

∂T

∂z=

k

ρcp

(∂2T

∂r2+

1

r

∂T

∂r

)+

Q

ρcp(T − T∞), (8.5)

u∂a

∂r+ w

∂a

∂z= DA

(∂2a

∂r2+

1

r

∂a

∂r

)− krab

2,

u∂b

∂r+ w

∂b

∂z= DB

(∂2b

∂r2+

1

r

∂b

∂r

)+ krab

2. (8.6)

117

at r = R, u = 0 w = we =U0zl,

DA∂a∂r

= ksa,∂T∂r

= −hsT, DB∂b∂r

= −ksa,

asr → ∞, w → 0, T → T∞, a→ a0, b→ 0,

(8.7)

Using

η =

√U0

νl

(r2 −R2

2R

), w =

U0z

lf ′ (η) , u = −

√νU0

l

R

rf (η) ,

θ (η) =T − T∞T∞

, ζ(η) =a

a0, h(η) =

b

a0, (8.8)

equation (8.6) is identically satisfied while Eqs. (8.7) to (8.10) yield

(1 + 2γη) (1 +M1) f′′′ + ff ′′ − (f ′)

2+ 2γ(1 +M1)f

′′

− 4

3(1 + 2γη)M1γλ1f

′′ − (1 + 2γη)2 λ1M1f′′2f ′′′ −Ha2f ′ = 0, (8.9)

(1 + 2γη) θ′′ + 2γθ′ + Pr(fθ′ + δθ) = 0, (8.10)

1

Sc((1 + 2γη)ζ ′′ + γζ ′) + fζ ′ −Kζh2 = 0, (8.11)

δ1Sc

((1 + 2γη)h′′ + γh′) + fh′ +Kζh2 = 0, (8.12)

f (0) = 0, f ′ (0) = 1, θ′ (0) = −α2 (1 + θ (0)) , f ′ (∞) = 0, θ (∞) = 0,

ζ ′ (0) = Ksζ(0), ζ (∞) → 1, δ1h′(0) = −Ksζ(0), h(∞) → 0.

(8.13)

118

The parameters are defined as follows:

γ =

(νl

U0R2

) 12

, Pr =µcpk, M1 =

1

µβ3c1, λ1 =

U30 z

2

2l3c21ν, K =

kra20l

U0

, δ1 =DB

DA

δ =Ql

ρcpU0

, Sc =ν

DA

, Ks =ksDA

√ν

U0

l, α2 = hs

√ν

U0

l, Ha2 =σ1β

20 l

ρU0

. (8.14)

The diffusion coefficients DA and DB are assumed same i. e. δ1 = 1

h(η) + ζ(η) = 1. (8.15)

Thus Eqs. (8.14), (8.15) and (8.17) become

(1 + 2γη)ζ ′′ + γζ ′ + Scfζ ′ − ScKζ(1− ζ)2 = 0, (8.16)

ζ ′(0) = Ksζ(0), ζ (∞) → 1. (8.17)

Local Nusselt number and skin friction are

Cf =τwρw2

e

, Nuz =zqw

k(T − T∞), (8.18)

τw =

[(µ+

1

βc

)(∂w

∂r

)− 1

6βc3

(∂w

∂r

)3]r=R

, qw = −k(∂T

∂r

)r=R

. (8.19)

CfRe1/2z = (1 +M1) f

′′ (0)− λ13M1f

′′3 (0) , NuzRe−1/2z = α2

(1 +

1

θ (0)

), (8.20)

where Rez = wezν

denotes the local Reynolds number.

119

8.2 Homotopic results

We select

f0 (η) = 1−exp (−η) , θ0 (η) =α

1− αexp (−η) , ζ0 (η) = 1−1

2exp (−Ksη) , (8.21)

Lf (η) =d3f

dη3− df

dη, Lθ (η) =

d2θ

dη2− θ, Lζ (η) =

d2ζ

dη2− ζ, (8.22)

Lf [A1 + A2 exp(η) + A3 exp(−η)] = 0, (8.23)

Lθ [A4 exp(η) + A5 exp(−η)] = 0, (8.24)

Lζ [A6 exp(η) + A7 exp(−η)] = 0, (8.25)

Ai (i = 1− 7) the arbitrary constants.

Convergence domain of the outcomes rely on the auxiliary variable ~. Fig. 8.2 displays

~-curves. Domain of convergence for auxiliary variables ~f , ~θ and ~ζ is −1.5 ≤ ~f ≤

−0.1, −2.5 ≤ ~θ ≤ −0.1 and −1.9 ≤ ~ζ ≤ −0.8 respectively.

8.3 Discussion

Dimensionless parameters Ha = M1 = α2 = δ = γ = 0.1, Pr = 0.9, K = 0.4,

Ks = 1.2 and Sc = 1.5 are fixed except the parameter as defined in figures. Fig. 8.3 is

plotted via Ha for the velocity outline. The velocity outline declines as the values of Ha

is increased. Larger Ha tends to higher Lorentz force. More resistive force develops via

larger Lorentz force and velocity outline of liquid declines. Fig. 8.4 addresses via γ for

the velocity outline. Larger γ first declines and then enhances the velocity outline. Larger

120

γ, R of cylinder declines and resistive force reduces for liquid and the velocity outline

enhances. Large M1 enhance the velocity outline (see Fig. 8.5). The liquid becomes less

viscous via larger M1.

Fig. 8.6 elucidatesHa effects on temperature θ(η). Temperature of liquid increases via

larger Ha. The temperature profile is enhanced for larger values of curvature parameter

(see Fig. 8.7). Temperature outline declines via M1 (see Fig. 8.8). Fig. 8.9 addresses

the plots of Pr for temperature profile θ(η). Temperature outline via Pr declines. In fact

larger Pr tends to small thermal diffusivity. Fig. 8.10 is plotted via conjugate parameter

α2 for temperature outline. Here variable α2 directly relates with temperature. Larger α2

tends to enhance heat transfer coefficient and temperature outline advances. Fig. 8.11

is sketched for temperature profile θ(η) via δ. Temperature outlines enhanced via heat

generation.

Fig. 8.12 presents the concentration outline viaHa. The concentration outline declines

via larger Ha. Fig. 8.13 reveals the concentration outline via γ. The concentration outline

first declines and then advances via larger γ. Fig. 8.14 is sketched viaM1 on concentration

outline ζ(η). ζ(η) outcome enhances via M1. Fig. 8.15 shows the influence of K on ζ(η).

ζ(η) declines via larger K. Larger Ks enhances ζ(η) (see Fig. 8.16). Fig. 8.17 addresses

the curves of ζ(η) via Sc. Concentration outline enhances via larger Sc.

Table 8.1 represents the convergence analysis. 25th orders need for convergence. Table

8.2 addresses validation of f ′′(0) via γ = 0 and Ha = 0 with published data. The out-

comes are matched. Values of f ′′(0) declines via M1 and advances via larger λ1. Table

8.3 reveals validation of CfRe1/2z when γ = 0 and Ha = 0. The outcomes are matched.

CfRe1/2z result improves via M1 and it decreases via λ1. Table 8.4 depicts Nusselt number

via variables. Heat transfer improves via larger γ, M1, Pr and α2 while it decreases with

121

Ha and δ. The heat transfer rate is via curvature parameter γ and liquid parameter M1.

Cylindrical shape tools via large γ found high magnitude of heat transfer. Table 8.5 ad-

dresses the skin friction via various parameters. The skin friction improves via γ and Ha

and reduces via M1. Table 8.6 addresses validation of f ′′(0) with [108, 109].

122

hf, hθ, hζ

f’’

(0),

θ’(

0),ζ

’(0)

-2 -1 0 1-4

-2

0

2

4

f ’’ (0)θ ’(0)ζ ’(0)

Fig. 8.2: ~-curves.

η

f’

(η)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ha = 0.5Ha = 0.7Ha = 0.9

Fig. 8.3: Plots via Ha for f ′(η).

123

η

f’(

η)

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ = 0.0γ = 0.1γ = 0.2

Fig. 8.4: Plots via γ for f ′(η).

η

f’

(η)

0 5 10 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M1 = 0.0M1 = 0.5M1 = 0.9

Fig. 8.5: Plots via M1 for f ′(η).

124

η

θ(η

)

0 2 4 6 8 10

0.05

0.1

0.15

0.2

Ha = 0.0Ha = 0.3Ha = 0.7

Fig. 8.6: Plots via Ha for θ(η).

η

θ(η

)

0 2 4 6 8 100

0.025

0.05

0.075

0.1

0.125

0.15

0.175

γ = 0.0γ = 0.2γ = 0.3

Fig. 8.7: Plots via γ for θ(η).

125

η

θ(η

)

0 2 4 6 8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

M1 = 0.0M1 = 0.5M1 = 0.9

Fig. 8.8: Plots via M1 for θ(η).

η

θ(η

)

0 2 4 6 8 10

0.05

0.1

0.15

0.2

Pr = 0.8Pr = 1.2Pr = 1.5

Fig. 8.9: Plots via Pr for θ(η).

126

η

θ(η

)

0 2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

α2 = 0.2α2 = 0.3α2 = 0.4

Fig. 8.10: Plots via α2 for θ(η).

η

θ(η

)

0 2 4 6 8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

δ = 0.0δ = 0.1δ = 0.2

Fig. 8.11: Plots via δ for θ(η).

127

η

ζ(η

)

0 10 20 30 40 50 600.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Ha = 0.0Ha = 0.5Ha = 0.9

Fig. 8.12: Plots via Ha for ζ(η).

η

ζ(η

)

0 10 20 30 40 50 60

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

γ = 0.0γ = 0.1γ = 0.3

Fig. 8.13: Plots via γ for ζ(η).

128

η

ζ(η

)

0 10 20 30 40 50 60

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

M1 = 0.0M1 = 0.5M1 = 0.9

Fig. 8.14: Plots via M1 for ζ(η).

η

ζ(η

)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K = 0.5K = 0.7K = 0.9

Fig. 8.15: Plots via K for ζ(η).

129

η

ζ(η

)

0 5 10 15 200.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ks = 0.3Ks = 0.7Ks = 1.2

Fig. 8.16: Plots via Ks for ζ(η).

η

ζ(η

)

0 10 20 30 40 50 600.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sc = 0.3Sc = 0.8Sc = 1.5

Fig. 8.17: Plots via Sc for ζ(η).

130

Table 8.1: Convergence via solutions at different order by fixing γ = M1 = λ1 = Ha =α2 = δ = 0.1, K = 0.6, Ks = 1.0, Sc = 1.2, and Pr = 0.7.

Estimations order −f ′′(0) −θ′(0) ζ ′(0)1 1.0222 0.11494 0.050305 1.0061 0.11970 0.0567110 1.0060 0.12078 0.0659115 1.0060 0.12108 0.0759420 1.0060 0.12108 0.0759425 1.0060 0.12108 0.07594

Table 8.2: Validation of f ′′(0) [107] via γ = Ha = 0. The present results are closed inbrackets.

λ1/M1 0.0 0.2 0.4 0.6 0.8 1.00.0 -1 -0.9131 -0.8452 -0.7906 -0.7454 -0.7071

(-1) (-0.91287) (-0.84516) (-0.79057) (-0.74536) (-0.70711)0.1 -1 -0.9159 -0.8493 -0.7950 -0.7498 -0.7114

(-1) (-0.91590) (-0.84929) (-0.79503) (-0.74979) (-0.71137)0.2 -1 -0.9190 -0.8536 -0.7997 -0.7544 -0.7158

(-1) (-0.91900) (-0.85358) (-0.79968) (-0.75442) (-0.71584)0.3 -1 -0.9222 -0.8580 -0.8045 -0.7593 -0.7205

(-1) (-0.92218) (-0.85804) (-0.80453) (-0.75927) (-0.72048)0.4 -1 -0.9254 -0.8627 -0.8096 -0.7644 -0.7254

(-1) (-0.92543) (-0.86267) (-0.80960) (-0.76436) (-0.72538)0.5 -1 -0.9288 -0.8675 -0.8149 -0.7697 -0.7305

(-1) (-0.92878) (-0.86749) (-0.81493) (-0.76971) (-0.73053)0.6 -1 -0.9322 -0.8725 -0.8205 -0.7754 -0.7360

(-1) (-0.93221) (-0.87252) (-0.82053) (-0.77534) (-0.73598)0.7 -1 -0.9357 -0.878 -0.8264 -0.7813 -0.7418

(-1) (-0.93574) (-0.87777) (-0.82643) (-0.78133) (-0.74174)0.8 -1 -0.9394 -0.8833 -0.8327 -0.7877 -0.7479

(-1) (-0.93938) (-0.88327) (-0.83267) (-0.78768) (-0.74788)0.9 -1 -0.9431 -0.8891 -0.8393 -0.7954 -0.7544

(-1) (-0.94312) (-0.88905) (-0.83930) (-0.79446) (-0.75443)1.0 -1 -0.9470 -0.8951 -0.8464 -0.8017 -0.7615

(-1) (-0.94698) (-0.89513) (-0.84637) (-0.80172) (-0.76145)

131

Table 8.3: Validation of skin friction coefficient Re1/2z Cf [107] when γ =Ha = 0. Thepresent results are in brackets

λ1/M1 0.0 0.2 0.4 0.6 0.8 1.00.0 -1 -1.0954 -1.1832 -1.2649 -1.3416 -1.4142

(-1) (-1.09545) (-1.18322) (-1.26491) (-1.34164) (-1.41421)0.1 -1 -1.0940 -1.1808 -1.2620 -1.3384 -1.4107

(-1) (-1.09395) (-1.18084) (-1.26199) (-1.33838) (-1.41073)0.2 -1 -1.0924 -1.1784 -1.2590 -1.3351 -1.4072

(-1) (-1.09245) (-1.17843) (-1.25902) (-1.33506) (-1.40718)0.3 -1 -1.0909 -1.1776 -1.2560 -1.3317 -1.4036

(-1) (-1.09092) (-1.17598) (-1.25600) (-1.33167) (-1.40356)0.4 -1 -1.0894 -1.1735 -1.2529 -1.3282 -1.3999

(-1) (-1.08938) (-1.17349) (-1.25291) (-1.32821) (-1.3999)0.5 -1 -1.0878 -1.1710 -1.2498 -1.3247 -1.3961

(-1) (-1.08782) (-1.17096) (-1.24976) (-1.32467) (-1.39609)0.6 -1 -1.0862 -1.1684 -1.2466 -1.3211 -1.3922

(-1) (-1.08625) (-1.16838) (-1.24655) (-1.32106) (-1.39223)0.7 -1 -1.0847 -1.1658 -1.2433 -1.3174 -1.3883

(-1) (-1.08465) (-1.16575) (-1.24327) (-1.31736) (-1.38827)0.8 -1 -1.0830 -1.1631 -1.2399 -1.3136 -1.3842

(-1) (-1.08304) (-1.16308) (-1.23991) (-1.31357) (-1.38422)0.9 -1 -1.0814 -1.1603 -1.2365 -1.3097 -1.3801

(-1) (-1.08141) (-1.16035) (-1.23646) (-1.30968) (-1.38006)1.0 -1 -1.0798 -1.1576 -1.2329 -1.3057 -1.3758

(-1) (-1.07975) (-1.15756) (-1.23293) (-1.30569) (-1.37578)

Table 8.4: Nusselt number via different variables.

α2 M1 γ Ha δ Pr NuzRe−1/2z

0.1 0.3 0.2 0.1 0.2 1.2 0.57120.3 0.58270.5 0.61400.1 0.0 0.2 0.1 0.2 1.2 0.3687

0.1 0.39680.4 0.3990

0.1 0.3 0.3 0.1 0.2 1.2 0.57510.5 0.58410.7 0.6138

0.1 0.3 0.2 0.1 0.2 1.2 0.42080.1 0.38520.3 0.3605

0.1 0.3 0.2 0.1 0.0 1.2 0.43070.1 0.42750.2 0.4119

0.1 0.3 0.2 0.1 0.1 0.8 0.38431 0.41151.2 0.4407

132

Table 8.5: CfRe1/2z via variables.

M1 Ha γ CfRe1/2z

0.0 0.1 0.2 0.0503180.1 0.0483370.3 0.0450140.3 0.0 0.2 0.036821

0.1 0.0450140.3 0.052441

0.3 0.1 0.0 0.0367190.1 0.0410620.2 0.045014

Table 8.6: Validation of f ′′(0) when γ =λ1= M1 = 0.

[108] [109]Ha HPM MHPM Exact solution Exact solution PR0 1 1 1 1 10.5 -1.1180 -1.1180341 -1.41421 -1.41421 -1.41421 -1.4142325 -2.44948 -2.44948 -2.44948 -2.449474

8.4 Main findings

Diffusion species in Powell-Eyring liquid flow via a stretched cylinder is discussed.

Heat generation/absorption and Newtonian heating are examined. Major findings:

• Larger M1 enhance the velocity outline and the temperature outline decreases.

• The temperature outline declines near the cylinder via lager γ and enhances via

larger Ha.

• Larger K decline the concentration outline and Ks enhances the concentration out-

line.

• Skin friction improves via M1 and controls λ1.

133

Chapter 9

Joule heating and viscous dissipation in chemical reactive flow

Viscous dissipation in MHD second grade material flow via a stretched cylinder is ex-

amined. Joule heating and Newtonian effects are discussed. Homogeneous/heterogeneous

reactions are studied. The outcomes are dealts via HAM. Temperature, velocity, skin fric-

tion coefficient and Nusselt number are studied in detail.

9.1 Formulation

Flow induced here is by an incompressible stretching cylinder. Analysis has been

carried out in the presence of Newtonian heating, dissipation and Joule heating. liquid

is conducting via constant magnetic field. Induced magnetic Reynolds number for small

induced magnetic field is not taken into account. For homogeneous reaction with cubic

autocatalysis is

A+ 2B → 3B, rate = krab2. (9.1)

On catalyst surface the first-order isothermal reaction is

134

A → B, rate = ksa. (9.2)

The relevant problems are

∂ (ru)

∂r+∂ (rw)

∂z= 0, (9.3)

u∂w

∂r+ w

∂w

∂z=ν

(∂2w

∂r2+

1

r

∂w

∂r

)+α∗1

ρ

u∂3w∂r3

+ w ∂3w∂z∂r2

− ∂2u∂r2

∂w∂r

+ ∂w∂z

∂2w∂r2

+1r(u∂2w

∂r2+ w ∂2w

∂z∂r− ∂u

∂r∂w∂r

+ ∂u∂r

∂w∂r)

− σ1β

20

ρw, (9.4)

u∂T

∂r+ w

∂T

∂z=α∗

(∂2T

∂r2+

1

r

∂T

∂r

)+v

cp

(∂w

∂r

)2

+α∗1

ρcp

(u∂w

∂r

∂2w

∂r2+ w

∂w

∂r

∂2w

∂z∂r

)+σβ2

0

ρcpw2, (9.5)

u∂a

∂r+ w

∂a

∂z= DA

(∂2a

∂r2+

1

r

∂a

∂r

)− krab

2,

u∂b

∂r+ w

∂b

∂z= DB

(∂2b

∂r2+

1

r

∂b

∂r

)+ krab

2, (9.6)

with constrains defined in Eq. 8.7. Applying

η =

√U0

νl

(r2 −R2

2R

), w =

U0z

lf ′ (η) , u = −

√νU0

l

R

rf (η) ,

θ (η) =T − T∞T∞

, ζ(η) =a

a0, h(η) =

b

a0. (9.7)

135

Eq. (9.3) is identically fulfilled and Eqs. (9.4− 9.7) are reduced to

(1+2γη)f ′′′ + ff ′′ − (f ′)2 + 2γf ′′ + 4γω(f ′f ′′ − ff′′′)

+ ω(1 + 2γη)(2f ′f′′′+ (f

′′)2 − ff

iv

)−Ha2f ′ = 0, (9.8)

(1+2γη)θ′′ + 2γθ′ + Pr fθ′ − PrEcωγf(f′′)2 + PrEc (1 + 2γη) ((f

′′)2

− ωff′′f

′′′+ ωf ′(f

′′)2)) + PrEcHa2(f

′)2 = 0, (9.9)

(1+2γη)ζ ′′ + 2γζ ′ + Scfζ ′ − ScKζ(1− ζ)2 = 0, (9.10)

f(0) = 0, f ′(0) = 1, θ′ (0) = −α2(1 + θ (0)), ζ

′(0) = Ks ζ(0),

f ′(∞) = 0, θ (∞) = 0, ζ(∞) = 1.

(9.11)

These parameters are defined as follows:

γ =

(νl

U0R2

) 12

, Pr =v

α∗ , Ec =U20

cp△T, K =

kra20l

we

, ω =α∗1U0

ρvl

Sc =ν

DA

, Ks =ksDA

√νl

U0

, Ha2 =σ1β

20 l

ρU0

, α2 = hs

√ν

U0

l. (9.12)

Expressions for skin friction and local Nusselt number are

Cf =2τwρw2

e

, Nuz =zqw

k(T − T∞), (9.13)

τw =

[µ∂w

∂r+ α1

(u∂2w

∂r2+ w

∂2w

∂z∂r− ∂u

∂r

∂w

∂r+∂u

∂r

∂w

∂r

)]r=R

, qw = −k(∂T

∂r

)r=R

,

(9.14)

136

in dimensionless variables one has

1

2CfRe

1/2z = (1 + 3ω)f ′′ (0) , NuzRe

−1/2z = α2

(1 +

1

θ (0)

). (9.15)

9.2 Homotopy results

We select ((f0, θ0, ζ0) and ((Lf ,Lθ,Lζ)) in the forms:

f0 (η) = (1− exp (−η)), θ0 (η) = (α2

1− α2

) exp (−η) , ζ0 (η) = (1− 1

2exp (−Ksη)),

(9.16)

Lf (f) =d3f

dη3− df

dη, Lθ (θ) =

d2θ

dη2− θ, Lζ (ζ) =

d2ζ

dη2− ζ, (9.17)

Lf [A1 + A2 exp(η) + A3 exp(−η)] = 0, (9.18)

Lθ [A4 exp(η) + A5 exp(−η)] = 0, (9.19)

Lζ [A6 exp(η) + A7 exp(−η)] = 0, (9.20)

where Ai the arbitrary constants.

9.2.1 Convergence analysis

The plots for ~-curves are shown in Fig. 9.1. Permissible values of ~f , ~θ and ~ζ are

−1.5 ≤ ~f ≤ −0.1, −2.5 ≤ ~θ ≤ −0.1 and −1.9 ≤ ~ζ ≤ −0.8.

9.3 Interpretation

Here velocity, temperature and concentration are discussed. Dimensionless variables

Ha = γ = ω = 0.1, Pr = 0.8, α2 = Ec = 0.1, K = 0.4, Ks = 1.2 and Sc = 1.5

137

hf , hθ , hζ

f’’(0

),

θ’(0

),

ζ’(0

)

-3 -2 -1 0-2

-1

0

1

f’’(0)θ’(0)ζ’(0)

Fig. 9.1: ~-curves

are fixed except the variable in figures. Curvature (γ) effect on flow is presented in Fig.

9.2. The velocity outline is enhanced via γ. Radius of cylinder is reduced via larger γ.

Fig. 9.3 illustrates the velocity outline via ω. Velocity and corresponding layer thickness

are increasing function of ω. The fluid velocity enhances via ω. It is in view of decay in

viscosity.

Fig. 9.4 shows the temperature variations via γ. It is noticed that thermal near and

away from cylinder is different. Further temperature is enhanced by convection. Fig. 9.5

is prepared for variation of temperature via ω. Temperature and thermal layer thickness

are decreasing function of ω. Clearly there is decrease in viscosity via ω. As a result

liquid moment enhances and less amount of heat is generated. θ(η) for α2 is shown in Fig.

9.6. There is an increase in thermal layer thickness via α2. Heat transfer coefficient also

enhances. Fig. 9.7 depicts variation of Pr on temperature. Temperature and thermal layer

thickness decay for Pr.

Fig. 9.8 displays concentration via γ. ζ(η) increases near the surface and declines

138

away from cylinder when γ enhanced. Fig. 9.9. shows variation of strength of K on

concentration profile. Higher K decays concentration. Curves of Sc and Ks on ζ(η) are

shown in Figs. 9.10− 9.11. Clearly concentration increases for Sc and Ks.

Table 9.1 is arranged for convergence analysis. Tabulated data depicts that 20th order of

approximations is sufficient. Table 9.2 addresses validation [110]. Outcomes are matched

exactly. Table 9.3 consists of numerical values of skin friction coefficient. Obviously

CfR1/2z increases via larger γ, Ha and ω. Table 9.4 has numerical values of Nussetl num-

ber. Nusselt number increases via γ, ω, Pr, α2 and it decreases forHa andEc. Tables (9.3

and 9.4) are numerical and HAM results for validation. Numerical outcomes of CfR1/2e

and NzR1/2e in Tables 9.3 and 9.4 are addressed correspondingly via shooting technique.

The outcomes are in favorable in these tables.

139

η

f’

(η)

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ = 0.0γ = 0.1γ = 0.2γ = 0.3

Fig. 9.2: Plots via γ for f ′(η).

η

f’

(η)

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ω = 0.1ω = 0.3ω = 0.5ω = 0.7

Fig. 9.3: Plots via ω for f ′(η).

140

η

θ(η

)

0 5 10

0.05

0.1

0.15

0.2

0.25

γ = 0.1γ = 0.2γ = 0.3γ = 0.4

Fig. 9.4: Plots via γ for θ(η).

η

θ(η

)

0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

ω = 0.1ω = 0.3ω = 0.5ω = 0.7

Fig. 9.5: Plots via ω for θ(η).

141

η

θ(η

)

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

α = 0.0α = 0.1α = 0.2α = 0.3

Fig. 9.6: Plots via α2 for θ(η).

η

θ(η

)

0 2 4 6 8 10

0.05

0.1

0.15

0.2

0.25

Pr = 0.8Pr = 1.0Pr = 1.2Pr = 1.5

Fig. 9.7: Plots via Pr for ζ(η).

142

η

ζ(η

)

0 10 20 30 40 50

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

γ = 0.0γ = 0.3γ = 0.6γ = 0.8

Fig. 9.8: Plots via γ for ζ(η).

η

ζ(η

)

0 10 20 30 40

0.5

0.6

0.7

0.8

0.9

1

K = 0.0K = 0.3K = 0.5K = 0.7

Fig. 9.9: Plots via K for ζ(η).

143

η

ζ(η

)

0 2 4 6 8 10 12 140.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ks = 0.7Ks = 0.9Ks = 1.2Ks = 1.5

Fig. 9.10: Plots via Ksfor ζ(η).

η

ζ(η

)

0 2 4 6 8 10 12 14

0.4

0.5

0.6

0.7

0.8

0.9

1

Sc = 0.8Sc = 1.0Sc = 1.2Sc = 1.5

Fig. 9.11: Plots via Sc for ζ(η).

144

Table 9.1: Series solutions convergence when γ = ω = 0.1, Ks = 1.2, Ec = 0.1,Sc = 0.9, α2 = Ha = 0.1, K = 0.4 and Pr = 0.8.

Estimations order −f ′′(0) −θ′(0) ζ ′(0)1 0.99267 0.12156 0.102945 0.98518 0.12674 0.1251010 0.98495 0.12694 0.1396615 0.98494 0.12695 0.1448820 0.98493 0.12696 0.1465825 0.98493 0.12696 0.14658

Table 9.2: Comparison of f ′′(0) by varying Ha and putting γ = 0, ω = 0 [110].

Ha [110] PR0.0 1.00000 1.000000.2 1.09545 1.095450.5 1.22475 1.224751.0 1.41421 1.414351.2 1.48324 1.483511.5 1.58114 1.581162 1.73205 1.73207

Table 9.3: Skin friction via variable.

γ ω Ha HAM-results Shooting-results0.1 0.1 0.1 1.1808 1.18050.2 1.1815 1.18150.3 1.1826 1.18260.1 0.1 0.2 1.2275 1.2273

0.2 1.2736 1.27340.3 1.3548 1.3547

0.1 0.1 0.1 1.2212 1.22120.2 1.2275 1.22730.3 1.2454 1.2453

145

Table 9.4: Nusselt number via different parameters.

γ ω Ha Ec Pr α2 HAM-results Shooting-results0.1 0.1 0.1 0.1 1.2 0.1 0.48296 0.482930.2 0.49482 0.494810.5 0.54176 0.541770.1 0.1 0.1 0.2 1.2 0.1 0.38361 0.38361

0.2 0.39363 0.393650.4 0.40741 0.40745

0.1 0.1 0.0 0.1 1.2 0.1 0.49782 0.497820.1 0.49480 0.494810.2 0.48613 0.48612

0.1 0.1 0.1 0.0 1.2 0.1 0.70225 0.702280.1 0.49482 0.494810.2 0.39367 0.39365

0.1 0.3 0.1 0.2 0.8 0.1 0.36591 0.365971 0.38834 0.388371.2 0.40420 0.40421

0.1 0.1 0.1 0.2 1.2 0.1 0.17217 0.172160.2 0.17713 0.177120.3 0.17883 0.17883

9.4 Final remarks

Major observations here include:

• Velocity via curvature and viscoelastic parameters is enhanced.

• Temperature and velocity for viscoelastic parameter have opposite results.

• Temperature is similar for conjugate and Eckert numbers.

• Concentration has opposite trend for homogenous and heterogeneous reactions.

• Heat transfer for viscoelastic, Prandtl and conjugate parameters are similar.

146

Chapter 10

Diffusion species in viscoelastic liquid flow with melting heat

The communication in this chapter inspects diffusion species in viscoelastic liquid flow.

Flow of liquid is via stretched cylinder with melting heat. Inclined magnetic field is used

electrified the liquid. Velocity outline, temperature outline and concentration outline are

eleborated via graphs. Outcoming equations are evaluated via HAM. Graphical outcomes

are explored via variables including in problem. CfRe1/2z and NzRe

−1/2z are computed.

10.1 Formulation

MHD viscoelastic liquid flow is examined via a stretched cylinder. Melting effect with

diffusion species are dealt. We take Tm∗ < T∞. Liquid is electrically conducting via

inclined magnetic field. Coordinates axes are shown via Fig. 8.1. The relevant problems

147

are

∂ (ru)

∂r+∂ (rw)

∂z= 0, (10.1)

u∂w

∂r+ w

∂w

∂z=ν

(∂2w

∂r2+

1

r

∂w

∂r

)+α∗1

ρ

u∂3w∂r3

+ w ∂3w∂z∂r2

− ∂2u∂r2

∂w∂r

+ ∂w∂z

∂2w∂r2

+1r(u∂2w

∂r2+ w ∂2w

∂z∂r− ∂u

∂r∂w∂r

+ ∂u∂r

∂w∂r)

− σβ2

0

ρSin2Φw, (10.2)

u∂T

∂r+ w

∂T

∂z=α∗

(∂2T

∂r2+

1

r

∂T

∂r

)+

Q

ρCp

(T − T∞), (10.3)

u∂a

∂r+ w

∂a

∂z=DA

(∂2a

∂r2+

1

r

∂a

∂r

)− krab

2,

u∂b

∂r+ w

∂b

∂z=DB

(∂2b

∂r2+

1

r

∂b

∂r

)+ krab

2. (10.4)

u = 0, w = we =

U0zl, T = Tm∗ , DB

∂b∂r

= −ksa, DA∂a∂r

= ksa, at r = R,

a→ a0, b→ 0, w → 0, T → T∞, as r → ∞,

(10.5)

k

(∂T

∂r

)= ρ (λ3 + Cs (Tm∗ − T0))u at r = R. (10.6)

Using

η =

√U0

νl

(r2 −R2

2R

), w =

U0z

lf ′ (η) , u = −

√νU0

l

R

rf (η) ,

θ (η) =T − TmT∞ − Tm

, ζ(η) =a

a0, h(η) =

b

a0. (10.7)

148

Eqs. (10.2− 10.5) are simplified and Eq. 10.1 fulfilled identically

(1 + 2γη) f ′′′ + ff ′′ − (f ′)2+ 2γf ′′ + 4γω(f ′f ′′ − ff

′′′)

+ ω (1 + 2γη) (2f ′f′′′+ (f

′′)2 − ff iv)−Ha2Sin2Φf ′ = 0, (10.8)

(1 + 2γη) θ′′ + 2γθ′ + Pr(fθ′ + δθ) = 0, (10.9)

(1 + 2γη) ζ ′′ + 2γζ ′ + Scfζ ′ − ScKζ(1− ζ)2 = 0, (10.10)

f ′(0) = 1, θ (0) = 0, P rf(0) +Mθ′ (0) = 0, ζ

′(0) = Ks ζ(0),

f ′(∞) = 0, θ (∞) = 1, ζ(∞) = 1.

(10.11)

The definition of parameters in above equations are

γ =

(νl

U0R2

) 12

, Pr =v

α∗ , K =kra

20l

U0

, ω =α∗1U0

ρvl, δ =

Ql

U0ρCp

,

Sc =ν

DA

, Ks =ksDA

√νl

U0

, Ha2 =σβ2

0 l

ρU0

, M =Cp(T∞ − Tm∗)

λ3 + Cs(Tm∗ − T0),(10.12)

local Nusselt number and skin friction are

Cf =τrzρw2

e

, Nuz =zqw

k(T∞ − Tm),

τw =

[µ∂w

∂r+ α1

(u∂2w

∂r2+ w

∂2w

∂z∂r+∂w

∂z

∂w

∂r− ∂u

∂r

∂w

∂r

)]r=R

, qw = −k(∂T

∂r

)r=R

,

(10.13)

1

2CfRe

1/2z = (1 + 3β1)f

′′ (0) , NuzRe−1/2z = −θ′

(0) , (10.14)

149

where Reynolds number Rez = wez/ν.

10.2 HAM outcomes

Select f0, θ0, ζ0 and Lf ,Lθ,Lζ as:

f0 (η) = (exp (−η)− M

Pr), θ0 (η) = 1− exp (−η) , ζ0 (η) = (1− 1

2exp (−Ksη))

(10.15)

Lf (η) =d3f

dη3− df

dη, Lθ (η) =

d2θ

dη2− θ, Lζ (η) =

d2ϕ

dη2− ζ, (10.16)

Lf [A1 + A2 exp(η) + A3 exp(−η)] = 0, (10.17)

Lθ [A4 exp(η) + A5 exp(−η)] = 0, (10.18)

Lζ [A6 exp(η) + A7 exp(−η)] = 0, (10.19)

Ai (i.e.i = 1− 7) the constants.

Convergence domain of outcomes depend upon ~ (see Fig. 10.1). The domain of auxiliary

variables are −2 ≤ ~f ≤ −0.7, −2 ≤ ~θ ≤ −0.8 and −2 ≤ ~ζ ≤ −0.7. Moreover the

results are converges of η (0 < η <∞) via ~f = ~θ = ~ζ = −0.9 (see Figs. 10.2− 10.4).

The square residual errors are defined below

∆fm =

∫ 1

0

[Rfm(η,f )]

2dη

∆θm =

∫ 1

0

[Rθm(η,θ )]

2dη

∆ζm =

∫ 1

0

[Rζm(η,ζ )]

2dη

150

hf , hθ , hζ

f’’(0

),

θ’(0

),

ζ’(0

)

-2 -1 0-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

f ’’ (0)θ ’ (0)ζ ’ (0)

Fig. 10.1: ~−curves

Table 10.1: Convergence for outcomes via γ = M = ω = Ha = δ = 0.1, K = 0.4,Ks = 1.2, Φ = π

4, Sc = 0.9and Pr = 1.2.

Order of estimations −f ′′(0) θ′(0) ϕ′(0)1 0.98975 0.84925 0.0881075 0.98612 0.77246 0.08143710 0.98388 0.75102 0.07956415 0.98230 0.74223 0.07882920 0.97949 0.73540 0.07824525 0.97949 0.73540 0.078245

151

10.3 Discussion

For outcomes the values of parameters γ = ω = Ha = Φ = δ = M = 0.1, Pr = 0.8,

K = 0.4, Ks = 1.0, Sc = 1.2 are fixed except the parameter appears in Figs. Fig.

10.5 addresses the velocity outline via γ. Velocity outline enhances via γ. Fig. 10.6 is

displayed via ω for velocity outline. The velocity outline increases via ω. Velocity outline

via Ha is depicted in Fig. 10.7. Larger Ha addresses higher the velocity outline. Large

M enhances the velocity outline (see Fig. 10.8). Fig. 10.9 addresses that higher values

of Φ tend to enhance liquid flow. It is noticed that Φ and magnetic field directly relate.

Behavior of γ via θ(η) is addressed in Fig. 10.10. Larger γ declines the temperature

outline. Fig. 10.11 addresses larger δ enhances the temperature outline. Fig. 10.12 reveals

that larger Pr tends to higher the temperature outlines. Plots of M for θ(η) is addressed

in Fig. 10.13. Lager M declines the tempreture outline. Higher values of K correspond to

decrease the concentration outline (see Fig. 10.14). ζ(η) viaKs is presented in Fig. 10.15.

The concentration outline enhances away from surface and opposite trend is noted near

the surface. Fig. 10.16 addresses that larger γ and M variables enhance the magnitude

of Nusselt number. Fig. 10.17 is plotted via Ha and Pr. Outcomes of Nusselt number

rise via Ha and Pr. Table 10.1 presents convergence of outcomes. 20th orders needs for

convergence of equations. Table 10.2 depicts that skin friction increases via γ, Ha and

ω. Table 10.3 witnesses that NuzRe−1/2z enhances via larger γ, ω, Ha, Pr and Φ while it

decreases through larger δ, M variables.

152

-1.4 -1.2 -1 -0.8 -0.6

Ñ

-0.00002

-0.00001

0

0.00001

0.00002

0.00003

0.00004

Dk

f

Fig. 10.2: ~−curves for residual error ∆fk .

-2 -1.5 -1 -0.5 0

Ñ

-3´10-7

-2´10-7

-1´10-7

0

1´10-7

2´10-7

3´10-7

4´10-7

Dk

Θ

Fig. 10.3: ~−curves for residual error ∆θk.

153

-1.4 -1.2 -1 -0.8 -0.6

Ñ

-0.00002

-0.00001

0

0.00001

0.00002

0.00003

0.00004

Dk

Ζ

Fig. 10.4: ~−curves for residual error ∆ζk.

η

f’

(η)

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ = 0.0γ = 0.3γ = 0.5γ = 0.7

Fig. 10.5: Plots via γ for f(η).

154

η

f’

(η)

2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ω = 0.0ω = 0.3ω = 0.5ω = 0.7

Fig. 10.6: Plots via ω for f(η).

η

f’

(η)

0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ha = 0.0Ha = 0.3Ha = 0.5Ha = 0.9

Fig. 10.7: Plots via Ha for f(η).

155

η

f’

(η)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 0.0M = 0.3M = 0.5M = 0.9

Fig. 10.8: Plots via M for f(η).

η

f’

(η)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Φ = 0.0Φ = π/4Φ = π/3Φ = π/2

Fig. 10.9: Plots via Φ for f(η).

156

η

θ(η

)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γ = 0.0γ = 0.3γ = 0.9γ = 1.7

Fig. 10.10: Plots via γ for θ(η).

η

θ(η

)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ = 0.0δ = 0.3δ = 0.5δ = 0.7

Fig. 10.11: Plots via δ for θ(η).

157

η

θ(η

)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr = 0.9Pr = 1.2Pr = 1.5Pr = 1.7

Fig. 10.12: Plots via Pr for θ(η).

η

θ(η

)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 0.0M = 0.3M = 0.5M = 0.7

Fig. 10.13: Plots via M for θ(η).

158

η

ζ(η

)

0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K = 0.0K = 0.3K = 0.5K = 0.7

Fig. 10.14: Plots via K for ζ(η).

η

ζ(η

)

0 5 10 15 20 25 30

0.5

0.6

0.7

0.8

0.9

1

Ks = 0.3Ks = 0.5Ks = 0.7Ks = 0.9

Fig. 10.15: Plots via Ks for ζ(η).

159

γ

Re

z-0.5

Nu

z

0 0.25 0.5 0.75 1

0.375

0.4

0.425

0.45

0.475

0.5

M = 0.1M = 0.5M = 0.9M = 1.2

Fig. 10.16: Plots for Nusselt number via γ and M .

Pr

Re z-0

.5N

u z

0.6 0.7 0.8 0.9

0.435

0.44

0.445

0.45

0.455

0.46

0.465

0.47

0.475

Ha = 0.1Ha = 0.5Ha = 0.9Ha = 1.2

Fig. 10.17: Plots for Nusselt number via γ and Pr.

160

Table 10.2: Skin friction via parameters.

γ ω Ha CfRe1/2z

0.0 0.1 0.1 1.09440.1 1.34270.2 1.60960.1 0.0 0.2 1.3175

0.1 1.34270.2 1.3547

0.1 0.1 0.0 1.23200.1 1.34270.2 1.4528

Table 10.3: Outcomes of Nusselt number via variables.

γ ω Ha M Pr δ Φ NuzRe−1/2z

0.0 0.1 0.1 0.1 1.2 0.1 0.1 0.459910.1 0.470990.5 0.481490.1 0.0 0.1 0.2 1.2 0.1 0.1 0.45965

0.1 0.470990.4 0.47962

0.1 0.1 0.0 0.1 1.2 0.1 0.1 0.498820.1 0.470990.2 0.44748

0.1 0.1 0.1 0.0 1.2 0.1 0.1 0.773380.1 0.736190.2 0.70235

0.1 0.3 0.2 0.8 0.1 0.1 0.319371 0.335961.2 0.47099

0.1 0.3 0.3 0.1 0.1 0.0 0.1 0.736190.1 0.693580.2 0.67218

0.1 0.3 0.3 0.1 0.1 0.1 0.0 0.70255π2

0.71338π4

0.73619

161

10.4 Final remarks

Major observations here include:

• Larger M presents opposite outline for the velocity and temperature.

• Outlines of temperature and velocity improve via larger γ.

• Concentration outline declines via larger K.

• NuzRe−12

z noted same outline via Ha and Pr.

162

Chapter 11

Influence of diffusion species in thermal an Oldroyd-B liquid flow

This chapter deals diffusion species in thermal flow of an Oldroyd-B liquid via mixed

conveciton. Mathematical equations are developed for velocity, temperature and concen-

tration functions through boundary layer theory. The resulting outcomes are dealt via

HAM. Graphical findings are addressed via variables.

11.1 Formulation

Oldroy-B liquid is examined via mix convection and stratification. Diffusion species

are accounted in stagnation flow. The heat develops via irreversible reaction is not dealt.

For homogeneous reaction with cubic autocatalysis is

A+ 2B → 3B, rate = krab2. (11.1)

On catalyst surface the first-order isothermal reaction is

A → B, rate = ksa. (11.2)

163

The relevant equations are

∂v

∂y+∂u

∂x= 0, (11.3)

v∂u

∂y+ u

∂u

∂x+ λ∗1

(v2∂2u

∂y2+ u2

∂2u

∂x2+ 2uv

∂2u

∂x∂y

)= ue

duedx

+ λ∗1u2e

d2uedx2

+ ν∂2u

∂y2

+ νλ∗2

{v∂3u

∂y3+ u

∂3u

∂x∂y2− ∂u

∂x

∂2u

∂y2− ∂u

∂y

∂2u

∂y2

}+ g1βT (T − T∞),

ρcp(u∂T

∂x+ v

∂T

∂y) = k

∂2T

∂y2, (11.4)

u∂a

∂x+ v

∂a

∂y= DA

∂2a

∂y2− krab

2,

u∂b

∂x+ v

∂b

∂y= DB

∂2b

∂y2+ krab

2. (11.5)

The corresponding conditions are presented in the forms:

u = uw(x) = cx, v = 0, T = Tw = T0 + b∗x,DA∂a

∂y= ksa,DB

∂b

∂y= −ksa at y = 0,

u = ue(x) = a∗x, T → T∞ = T0 + dx, a→ a0, b→ 0 as y → ∞. (11.6)

Using

u = cxf′(η), v = −

√cνf(η), η =

√c

νy, θ =

T − T∞Tw − T0

, ζ(η) =a

a0, h(η) =

b

a0.

(11.7)

Eq. (11.3) is identically fulfilled and Eqs. (11.4− 11.6) are reduced to

f ′′′ − f ′2 + ff ′′ + A2 + β1(2ff′f ′′ − f 2f ′′′) + β2(f

′′2 − ff iv) + λθ = 0, (11.8)

θ′′ − Pr(f ′θ − fθ′ + Sf ′) = 0, (11.9)

ζ ′′ + Scfζ ′ − ScKζ(1− ϕ)2 = 0, (11.10)

164

f(η) = 0, f ′(η) = 1, θ(η) = 1− S, ζ

′(η) = Ks ζ(η) at η = 0,

f ′(η) = A, θ(η) = 0, ζ(η) = 1 as η → ∞.

These parameters are defined as follows:

λ =Grx

Re2x, Grx =

g1βT (Tw − T∞)x3

ν2, Rex =

ue(x)x

ν, A =

a∗

c, Pr =

ν

α∗ ,

S =d

b∗, β1 = λ∗1c, β2 = λ∗2c, K =

k1a20

c, Sc =

ν

DA

. Ks =ksDA

√ν

c. (11.11)

Expressions for local Nusselt number is

NuxRe− 1

2 = −θ′(0). (11.12)

11.2 HAM outcomes

Select f0, θ0, ζ0 and Lf , Lθ,Lζ as:

θ0(η) = exp (−η) , f0(η) = Aη + (1− A)(1− exp (−η)), ζ0 (η) = (1− 1

2exp (−Ksη)),

(11.13)

Lf =d3f

dη3− df

dη, Lθ =

d2θ

dη2− θ, Lζ =

d2ζ

dη2− ζ, (11.14)

Lf [A1 + C2 exp(η) + A3 exp(−η)] = 0, Lθ [A4 exp(η) + A5 exp(−η)] = 0, (11.15)

Lζ [A6 exp(η) + A7 exp(−η)] = 0, (11.16)

fm(η) = f ∗m(η) + A1 + A2 exp(η) + A3 exp(−η), (11.17)

θm(η) = θ∗m(η) + A4 exp(η) + A5 exp(−η), (11.18)

ζm(η) = ζ∗m(η) + A6 exp(η) + A7 exp(−η), (11.19)

165

hf , hθ , hζ

f’’(0

),θ

’(0

),ζ

’(0

)

-3 -2 -1 0 1 2-4

-3

-2

-1

0

1

2

f’’(0)θ’(0)ζ’(0)

Fig. 11.1: ~−curves

The convergence domain are presented in Fig. 11.1 with ranges of auxilary variables as

~f , ~θ and ~ζ are −1.5 ≤ ~f ≤ −0.1, −2.5 ≤ ~θ ≤ −0.1 and −1.9 ≤ ~ζ ≤ −0.8.

Table 11.1: Convergence for the solutions via β1 = β2 = A = λ = S = 0.1, Pr = 1,K = 0.4. Ks = 0.9, Sc = 1.2 and ~ = −1.2.

Estimations order −f ′′(0) −θ′(0) ζ ′(0)

2 0.89416 0.95400 0.405726 0.90486 0.97746 0.35907

12 0.90876 0.98626 0.3416418 0.91064 0.98902 0.3370524 0.91064 0.99032 0.3352030 0.91064 0.99032 0.33520

11.3 Discussion

The values of dimensionless parameters for results are A = λ = S = β2 = β1 = 0.1,

Pr = 0.8, K = 0.7, Ks = 0.9 and Sc = 1.2. These variables are fixed except the variable

mentioned in Figures. The velocity outline via A is presented in Fig. 11.2. The velocity

outline enhances via A < 1 and declines via A > 1. The is no boundary layer at A = 1.

166

Velocity outline enhances via larger λ3 (see Fig. 11.3). Larger β1 declines velocity outline

(see Fig. 11.4). Fig. 11.5 displays the velocity outline via β2. The velocity enhances via

larger Dehorah number. Figs. 11.6 − 11.7 address the temperature outlines via S and Pr

respectively. The temperature declines via larger S and Pr. The concentration outline

enhances via β2 (see Fig. 11.8). Fig. 11.9 addresses the curves of K via concentration.

ζ(η) declines via larger K. Fig. 11.10 displays plots via Ks for concentration outline.

Larger Sc correspond higher concentration (see Fig. 11.11). Fig. 11.12 reflects that larger

Pr and S advance the Nusselt number. Fig. 11.13 is graphical outcomes for Nusselt

number via λ and S. Nusselt number outcomes decline via λ and depict higher values via

S. Table 11.1 presents the convergence of outcomes. Table 11.2 indicates validation of

f ′′(0) via λ = 0. Table 11.3 reflects validation of outcomes via HAM and Bvp4c. The

outcomes are matched via both methods.

167

η

f’(

η)

0 1 2 30.8

0.9

1

1.1

1.2

A = 0.8A = 0.9A = 1.0A = 1.1A = 1.17

Fig. 11.2: Plots via A for f ′(η).

η

f’(

η)

0 1 2 3 4 5 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ = 0.0λ = 0.3λ = 0.6λ = 0.9

Fig. 11.3: Plots via λ for f ′(η).

168

η

f’(

η)

0 1 2 3 4 5 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β1 = 0.0β1 = 0.3β1 = 0.6β1 = 0.9

Fig. 11.4: Plots via β1 for f ′(η).

η

f’(

η)

0 1 2 3 4 5 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

β2 = 0.0β2 = 0.3β2 = 0.6β2 = 0.9

Fig. 11.5: Plots via β2 for f ′(η).

169

η

θ(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S = 0.0S = 0.1S = 0.2S = 0.3

Fig. 11.6: Plots via S for θ(η).

η

θ(η

)

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pr = 0.1Pr = 0.3Pr = 0.7Pr = 1.2

Fig. 11.7: Plots via Pr for θ(η).

170

η

ζ(η

)

0 5 10 15 200.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

β2 = 0.0β2 = 0.3β2 = 0.7β2 = 0.9

Fig. 11.8: Plots via β2 for ζ(η).

η

ζ(η

)

0 5 10 15 20 25 300.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

K = 0.1K = 0.3K = 0.7K = 0.9

Fig. 11.9: Plots via K for ζ(η).

171

η

ζ(η

)

0 5 10 15

0.5

0.6

0.7

0.8

0.9

1

Ks = 0.7Ks = 0.9Ks = 1.2Ks = 1.5

Fig. 11.10: Plots via Ks for ζ(η).

η

ζ(η

)

0 2 4 6 8 10

0.4

0.5

0.6

0.7

0.8

0.9

1

Sc = 0.8Sc = 1.0Sc = 1.2Sc = 1.5

Fig. 11.11: Plots via Sc for ζ(η).

172

Pr

Re x-0

.5N

ux

0 0.25 0.5 0.75 1-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

S = 0.8S = 0.9S = 1.0S = 1.2

Fig. 11.12: Plots for Nusselt number via S and Pr.

S

Re x-0

.5N

ux

0 0.25 0.5 0.75 10.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

λ = 0.0λ = 0.3λ = 0.5λ = 0.9

Fig. 11.13: Plots for Nusselt number via λ and S.

173

Table 11.2: Comparison of f ′′(0) via β1, β2 and A when λ = 0 through the refs.[38, 114, 115, 113].

Newtonian fluid Maxwell fluid Oldroyd-B fluid(β2 = 0, β1 = 0) (β2 = 0.2, β1 = 0) (β2 = 0.2, β1 = 0.2)

A Ref. [114] Ref. [115] Ref. [38] Ref.[113] PR Ref. [38] Ref. [113] PR Ref. [38] Ref. [113] PR0.01 -0.9980 -0.9981 -0.9963 -0.9933 -1.0499 -1.0428 -1.0400 -0.9583 -0.9560 -0.95100.02 -0.9958 -0.9958 -0.9930 -0.9915 -1.0476 -1.0394 -1.0354 -0.9567 -0.9531 -0.95000.05 -0.9876 -0.9876 -0.9830 -0.9811 -1.0393 -1.0296 -1.0255 -0.9490 -0.9460 -0.94300.10 -0.9694 -0.9694 -0.9694 –0.9603 -0.9590 -1.0207 -1.0124 -1.0100 -0.9330 -0.9296 -0.92660.20 -0.9181 -0.9181 -0.9181 -0.9080 -0.9060 -0.9681 -0.9675 -0.9576 -0.8890 -0.8875 -0.88330.50 -0.6673 –0.6673 -0.6673 -0.6605 -0.6585 -0.7078 -0.7082 -0.6980 -0.6549 -0.6578 -0.65781.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00002.00 2.0175 2.0175 2.0175 2.0181 2.0181 2.2225 2.2453 2.2453 2.2255 2.2370 2.2370

Table 11.3: HAM outcomes and Bvp4c outcomes via λ = 0.1, β2 = 0.1, A = 0.1,S = 0.1, Pr = 0.8.

f ′(η) θ(η) ζ(η)η β1 = 0 β1 = 0.1 β1 = 0 β1 = 0.1 β1 = 0 β1 = 0.1

HAM Bvp4c HAM Bvp4c HAM Bvp44c HAM Bvp4c HAM Bvp4c HAM BVP4c0.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0000.5 0.5778 0.5750 0.6975 0.6909 0.5778 0.5750 0.5794 0.5750 0.5070 0.5027 0.5074 0.50741 0.5164 0.5175 0.6384 0.6310 0.5164 0.5175 0.5183 0.5117 0.7223 0.7220 0.7243 0.7242

1.5 0.4612 0.4651 0.5364 0.5343 0.4612 0.4651 0.4634 0.4651 0.7715 0.7710 0.7760 0.77602 0.4119 0.4131 0.4532 0.4503 0.4119 0.4131 0.4142 0.4131 0.8663 0.8640 0.8673 0.8670

2.5 0.3676 0.3669 0.4176 0.4131 0.3676 0.3669 0.3702 0.3790 0.9167 0.9161 0.9187 0.91853 0.2928 0.2936 0.3567 0.3553 0.2928 0.2936 0.3307 0.3332 0.9507 0.9500 0.9525 0.9525

3.5 0.2613 0.2642 0.3308 0.3332 0.2613 0.2642 0.2954 0.2936 0.9702 0.9692 0.9742 0.97404 0.2332 0.2339 0.2865 0.2846 0.2332 0.2339 0.2689 0.2642 0.9825 0.9803 0.9862 0.9862

4.5 0.2081 0.2079 0.2508 0.2532 0.2081 0.2079 0.2357 0.2339 0.9902 0.9892 0.9977 0.99765 0.1658 0.1641 0.2220 0.2262 0.1658 0.1641 0.1503 0.1504 0.9936 0.9920 0.9988 0.9987

5.5 0.1322 0.1319 0.1987 0.1996 0.1322 0.1319 0.1201 0.1205 0.9965 0.9950 0.9968 0.99686 0.1050 0.1050 0.1720 0.1720 0.1055 0.1055 0.1074 0.1074 0.9976 0.9960 0.9980 0.9980

11.4 Main findings

Major findings of this chapter are list here;

• Velocity outline increases via λ, β2 and opposite trend is noted via β1. Same outlines

of velocity and concentration are noted via Deborah number (through retardation

time).

• Larger stratified variable, Prandtl number correspond to reduction in temperature.

Magnitude of heat transfer enhances via Pr, S and it reduces via λ.

174

Chapter 12

Numerical simulation for chemical species and Joule heating in MHD

flow of Williamson fluid

Convective flow of Williamson fluid by a cylinder and flat sheet is addressed in this

chapter. Convective condition is used for heat transfer mechanism. The species of auto

catalyst and reactant are used to regulate the concentration. Convection or evaporation

for temperature phase change is analyzed through homogeneous and heterogeneous reac-

tions. The transformed differential systems is numerically evaluated. Impacts of pertinent

parameters of interest are graphically discussed. Comparison for cylinder and flat sheet

is arranged. The velocity and temperature profiles in case of cylinder is more dominated

when compared with flat sheet while reverse trend is noted in concentration profiles for

Williamson fluid. Skin friction and Nusselt number are accounted higher for cylinder than

flat sheet. Higher values of Weissenberg number reduce skin friction.

12.1 Formulation

The two-dimensional convective flow of Williamson liquid by a stretched cylinder is

assumed. The temperature of cylinder is regulated by convection process of hot fluid at

175

Tf . Heat generation/absorption, viscous dissipation and Joule heating are incorporated to

explore the heat transfer properties. Chemical species are used. The geometry of flow is

shown in Fig. 12.1.

Fig. 12.1: Schematic representation of problem.

The equations governing present flow are

∂ (rw)

∂z+∂ (ru)

∂r= 0, (12.1)

u∂w

∂r+ w

∂w

∂z= ν

(∂2w

∂r2+

1

r

∂w

∂r+√2Γ∂w

∂r

∂2w

∂r2+

Γ√2r

(∂w

∂r)2)− σ1β

20

ρw, (12.2)

w∂T

∂z+ u

∂T

∂r=α∗

r

∂

∂r

(r∂T

∂r

)+ν

cp

(∂w

∂r

)2

+σβ2

0

ρcpw2 +

Q

ρcp(T − T∞) , (12.3)

u∂a

∂r+ w

∂a

∂z= DA

(∂2a

∂r2+

1

r

∂a

∂r

)− krab

2,

u∂b

∂r+ w

∂b

∂z= DB

(∂2b

∂r2+

1

r

∂b

∂r

)+ krab

2, (12.4)

176

at r = R w = we =U0z

l, u = 0, DA

∂a

∂r= −aks, DB

∂b

∂r= aks,

− k

(∂T

∂r

)= hf (Tf − T ) ,

as r → ∞, w → 0, T → T∞, a→ a0, b→ 0. (12.5)

Here Γ > 0 the time constant.

Using transformations:

η =

√U0

νl

(r2 −R2

2R

), ψ =

√weνzRf (η) w =

U0z

lf ′ (η) ,

u =−√νU0

l

R

rf (η) , θ (η) =

T − T∞Tf − T∞

, ζ(η) =a

a0, h(η) =

b

a0, (12.6)

equation (12.3) is satisfied automatically and Eqs. (12.4− 12.7) become

(1 + 2γη)f ′′′ + ff ′′ − f ′2 + 2γf ′′ + 2ϵ (1 + 2γη)32 f ′′f ′′′

+ 3ϵγ (1 + 2γη)12 (f ′′)2 −Ha2f ′ = 0, (12.7)

(1 + 2γη)θ′′+ 2γθ′ + Prθ′f + PrEc(1 + 2γη)(f ′′)2

+ PrEcHa2(f ′)2 + Prδθ = 0, (12.8)

1

Sc((1 + 2γη)ζ ′′ + γζ ′) + fζ ′ −Kζh2 = 0, (12.9)

δ1Sc

((1 + 2γη)h′′ + γh′) + fh′ +Kζh2 = 0, (12.10)

177

with

θ′(0) = −Bi (1− θ(0)) , f(0) = 0, f ′(0) = 1, ζ ′(0) = Ksζ(0), δ1h

′(0) = −Ksζ(0),

θ (η) = 0, f ′(η) = 0, ζ(η) → 1, h(η) → 0 as η → ∞.

(12.11)

where γ = 1R

√νlU0

, ϵ = ΓU3/20 z

√2νl

32

, Ha2 = β20ρl

ρU0, Ec = U2

0

cp∆T, δ = Ql

ρcpU0,

Pr = να∗ , Bi = hf

k

√νlU0

, K =kra20l

U0, Ks = ks

DA

√νlU0

, Sc = νDA

and δ1 = DB

DA.

For same diffusion coefficients DA and DB, one has

ζ(η) + h(η) = 1.

Equations (12.11), (12.12) and (12.13) yield

(1 + 2γη)ζ ′′ + 2γζ ′ + Scfζ ′ − ScKζ(1− ζ)2 = 0. (12.12)

ζ ′(0) = Ksζ(0), ζ(η) → 1 as η → ∞. (12.13)

The skin friction and Nusselt number;

Cf =τw

ρU20 z

2

2l2

, Nuz =zqw

k∗(Tw − T∞), (12.14)

where τw and qw are

τw = µ

(∂w

∂r+

1√2Γ(∂w

∂r)2)

r=R

, qw = −k(∂T

∂r

)r=R

. (12.15)

178

Dimensionless variables finally yield

CfRe1/2z = f ′′ (0) + ϵf ′′2 (0) , NuzRe

−1/2z = −θ′ (0) , (12.16)

where Re1/2z =U

1/20 z

ν1/2l1/2denotes the local Reynolds number.

12.2 Implicit finite difference scheme

Implicit finite difference scheme [112] is utilized to solve Eqs.(12.9− 12.15). For this

purpose the first step is to reduce Eqs.(12.9 − 12.15) to first order and v, u, w, t, g, p and

q are new variables (u = f ′, v = u′, w = v′, t = g′, q = p′) and thus Eqs. (12.9− 12.15)

become

(1 + 2γη)v′ + 2γv + 3ϵ(1 + 2γη)1/2γv2 + 2ϵ(1 + 2γη)3/2vv′ + fv − u2 −Ha2u = 0,

(12.17)

(1 + 2γη)t′ + 2γt+ Pr tf + PrEc(1 + 2γη)v + PrEcHa2u2 + Prδg = 0, (12.18)

1

Sc((1 + 2γη)q′ + 2γq) + qf −Kp(1− p)2 = 0. (12.19)

For difference equations of Eqs.(12.9 − 12.15) in z − η plane ( see Fig. 12.2). The

points are:

z0 = 0, zi = zi−1 + ki, i = 1, 2, 3...I,

η0 = 0, zi = ηj−1 + hj, j = 1, 2, 3...J.

Here ki-hj the ∆z-∆η-spacing respectively. Centralizing via (zi, ηj−1/2)

179

f ij − f i

j−1

hj=uij + uij−1

2, (12.20)

uij − uij−1

hj=vij + vij−1

2, (12.21)

θij − gθij−1

hj=tij + tij−1

2, (12.22)

ζ ij − ζ ij−1

hj=qij + qij−1

2. (12.23)

180

Fig. 12.2: Net Keller box for finite difference approximation.

Using centering difference approximations about (zi−1/2, ηj−1/2) for Eqs.(12.19−12.21).

We have

(1 + 2γη)(vij − vij−1) + 2γhvij−1/2 + 3ϵ(1 + 2γη)1/2kh(vij−1/2)2 + hf i

j−1/2vij−1/2

+ 2ϵh(1 + 2γη)3/2vij−1/2wij−1/2 − h(uij−1/2)

2 −Ha2huij−1/2 = Lj−1/2, (12.24)

(1 + 2γη)(tij − tij−1) + 2γhtij−1/2 + Prhtij−1/2fij−1/2 + (1 + 2γη)PrEch((vij − vij−1)

2

+Ha2(uij − uij−1)2) + δPrh(gij − gij−1) =M∗

j−1/2, (12.25)

1

Sc((1 + 2γη)(qij − qij−1) + 2γhqij−1/2) + hqij−1/2f

ij−1/2 −Khpij−1/2(1− pij−1/2)

2

= Nj−1/2, (12.26)

181

with

Lj−1/2 = −((1 + 2γη)(vi−1j − vi−1

j−1) + 2γhvi−1j−1/2 + 3ϵ(1 + 2γη)1/2kh(vi−1

j−1/2)2

+ hf i−1j−1/2v

i−1j−1/2 + 2ϵh(1 + 2γη)3/2vi−1

j−1/2wi−1j−1/2 − h(ui−1

j−1/2)2 −M2hui−1

j−1/2),

M∗j−1/2 = −((1 + 2γη)(ti−1

j − ti−1j−1) + 2γhti−1

j−1/2 + Prhti−1j−1/2f

i−1j−1/2)

+ (1 + 2γη)PrEch((vi−1j − vi−1

j−1)2 +M2(ui−1

j − ui−1j−1)

2) + δPrh(ti−1j − ti−1

j−1),

Nj−1/2 = −(1

Sc((1 + 2γη)(qi−1

j − qi−1j−1) + 2γhqi−1

j−1/2) + hqi−1j−1/2f

i−1j−1/2)

− hKpi−1j−1/2(1− pi−1

j−1/2)2,

here the known variables are Ej−1/2, Lj−1/2 and Mj−1/2,

with

f i0 = 0, ui0 = 1, uiJ = 0, ti0 = −Bi(1− gi0), giJ = 0, qi0 = Kspi0), piJ = 0,

(12.27)

Newton’s method linearize the Eqs (12.26− 12.28) which are in nonlinear algebraic equa-

182

tions in the forms:

f(i+1)j = (f

(i)j + δ∗f

(i)j ),

u(i+1)j = (u

(i)j + δ∗u

(i)j ),

v(i+1)j = (v

(i)j + δ∗v

(i)j ),

g(i+1)j = (g

(i)j + δ∗g

(i)j ),

t(i+1)j = (t

(i)j + δ∗t

(i)j ),

p(i+1)j = (p

(i)j + δ∗p

(i)j ),

q(i+1)j = (q

(i)j + δ∗q

(i)j ). (12.28)

Using Eq. (12.30) in Eqs.(12.26− 12.28) and ignoring the higher order of δ∗, one obtains

(r1)j = δ∗fj − δ∗fj−1 − 0.5hj(δ∗uj + δ∗uj−1), (12.29)

(r2)j = δ∗uj − δ∗uj−1 − 0.5hj(δ∗vj + δ∗vj−1), (12.30)

(r3)j = δ∗gj − δ∗gj−1 − 0.5hj(δ∗tj + δ∗tj−1), (12.31)

(r4)j = δ∗pj − δ∗pj−1 − 0.5hj(δ∗qj + δ∗qj−1), (12.32)

183

(a∗1)j−1/2δ∗vj + (a∗2)j−1/2δ

∗vj−1 + (a∗3)j−1/2δ∗uj + (a∗4)j−1/2δ

∗uj−1

+ (a∗5)j−1/2δ∗fj + (a∗6)j−1/2δ

∗fj−1 = (r5)j−1/2, (12.33)

(b∗1)j−1/2δ∗tj + (b∗2)j−1/2δ

∗tj−1 + (b∗3)j−1/2δ∗fj + (b∗4)j−1/2δ

∗fj−1

+ (b∗5)j−1/2δ∗vj + (b∗6)j−1/2δ

∗vj−1 + (b∗7)j−1/2δ∗uj + (b∗8)j−1/2δ

∗uj−1+

(b∗9)j−1/2δ∗gj + (b∗10)j−1/2δ

∗gj−1 = (r6)j−1/2, (12.34)

(c∗1)j−1/2δ∗qj + (c∗2)j−1/2δ

∗qj−1 + (c∗3)j−1/2δ∗pj + (c∗4)j−1/2δ

∗pj−1

+ (c∗5)j−1/2δ∗fj + (c∗6)j−1/2δ

∗fj−1 = (r7)j−1/2, (12.35)

where

(a∗1)j−1/2 =((1 + 2γη) + hγ + 3ϵ(1 + 2γη)1/2khvj−1/2 + 2ϵ(1 + 2γη)3/2vj−1/2

+ ϵh(1 + 2γη)3/2wj−1/2 +hfj−1/2

2), (12.36)

(a∗2)j−1/2 =(−(1 + 2γη) + hγ + 3ϵ(1 + 2γη)1/2khvj−1/2 − 2ϵ(1 + 2γη)3/2vj−1/2

+ ϵh(1 + 2Kη)3/2wj−1/2 +hfj−1/2

2), (12.37)

(a∗3)j−1/2 =(h

2vj−1/2, (a∗4)j−1/2 = (a∗3)j−1/2), (12.38)

(a∗5)j−1/2 =(h[uj−1/2 −1

2M2]), (a∗6)j−1/2 = ((a5)j−1/2), (12.39)

(b∗1)j−1/2 =((1 + 2γη) + γh+Prhfj−1/2

2), (12.40)

(b∗2)j−1/2 =(−(1 + 2γη) + γh+Prhfj−1/2

2), (12.41)

(b∗3)j−1/2 =(Prhtj−1/2

2), (b∗4)j−1/2 = ((b∗3)j−1/2), (12.42)

(b∗5)j−1/2 =(hPrEcM2uj−1/2), (b∗6)j−1/2 = ((b∗5)j−1/2), (12.43)

(b∗7)j−1/2 =(Prδ

2), (b∗8)j−1/2 = ((b∗7)j−1/2), (12.44)

(b∗9)j−1/2 =(PrEchvj−1/2), (b∗10)j−1/2 = ((b∗9)j−1/2), (12.45)

184

(c1)j =(1 + 2γη) +fj− 1

2

2(γ + Sc

fj− 12

2), (12.46)

(c2)j =− (1 + 2γη) +fj− 1

2

2(γ + Sc

fj− 12

2), (12.47)

(c3)j =hj4Scpj− 1

2

2, (c4)j = (c3)j, (12.48)

(c5)j =− ScKhj2

+ 2ScKhj2qj− 1

2− ScK

hj2(qj− 1

2)2, (c6)j = (c5)j, (12.49)

(r5)j−1/2 =− h(1 + 2γη)wj−1/2 − 2γhvj−1/2 − 3ϵ(1 + 2γη)1/2khv2j−1/2

− 2ϵh(1 + 2γη)3/2vj−1/2wj−1/2 − hfj−1/2vj−1/2 + h(uj−1/2)2

−Ha2huj−1/2 + Lj−1/2, (12.50)

(r6)j−1/2 =− (1 + 2γη)tj + (1 + 2γη)tj−1 − 2γhtj−1/2 − Prhtj−1/2fj−1/2 +M∗j−1/2

− PrEch[(vj−1/2)2 +Ha2(uj−1/2)

2]− δPrgj−1/2 (12.51)

(r7)j−1/2 =− (1 + 2γη)(qj − qj−1)− 2γhqj−1/2 − Schqj−1/2fj−1/2

+ hScKpj−1/2(1− pj−1/2)2 +Nj−1/2, (12.52)

the conditions are

δ∗f0 = 0, δ∗u0 = 0, δ∗g0 = 0, δ∗p0 = 0, δ∗uJ = 0, δ∗tJ = 0, δ∗pJ = 0.

(12.53)

12.3 The block tridiagonal matrix

The linearized Eq.(12.30− 12.36) has a block tridiagonal structure in matrix form:

185

For J = 1

δ∗f1 − δ∗f0 −h12(δ∗u1 − δ∗u0) = (r1)1, (12.54)

δ∗u1 − δ∗u0 −h12(δ∗v1 − δ∗v0) = (r2)1, (12.55)

δ∗g1 − δ∗g0 −h12(δ∗t1 − δ∗t0) = (r3)1, (12.56)

δ∗q1 − δ∗q0 −h12(δ∗p1 − δ∗p0) = (r4)1 (12.57)

(a1)1δ∗v1 − (a2)1δ

∗v0 + (a3)1δ∗f1 − (a4)1δ

∗f0 + (a5)1δ∗u1 − (a6)1δ

∗u0 = (r5)1,

(12.58)

(b1)1δ∗t1 − (b2)1δ

∗t0 + (b3)1δ∗f1 − (b4)1δ

∗f0 + (b5)1δ∗u1 − (b6)1δ

∗u0

+ (b7)1δ∗s1 − (b8)1δ

∗s0 + (b9)1δ∗v1 − (b10)1δ

∗v0 = (r6)1, (12.59)

(c1)1δ∗p1 − (c2)1δ

∗p0 + (c3)1δ∗f1 − (c4)1δ

∗f0 + (c5)1δ∗q1 − (c6)1δ

∗q0 = (r7)1. (12.60)

186

In matrix form put d = −h1

2, we get

0 0 0 1 0 0 0

d 0 0 0 d 0 0

0 d 0 0 0 d 0

0 0 d 0 0 0 d

(a2)1 0 0 (a3)1 (a1)1 0 0

(b10)1 (b2)1 0 (b3)1 (b9)1 (b1)1 0

0 0 (c2)1 (c3)1 0 0 (c1)1

δ∗v0

δ∗t0

δ∗p0

δ∗f1

δ∗v1

δ∗t1

δ∗p1

+

d 0 0 0 0 0 0

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

(a5)1 0 0 0 0 0 0

(b5)1 (b7)1 0 0 0 0 0

0 0 (c5)1 0 0 0 0

δ∗u1

δ∗s1

δ∗q1

δ∗f2

δ∗v2

δ∗t2

δ∗p2

=

(r1)1

(r2)1

(r3)1

(r4)1

(r5)1

(r6)1

(r7)1

,

that is

[A1

] [δ∗1

]+

[C1

] [δ∗2

]=

[r1

]. (12.61)

187

For J = 2

δ∗f2 − δ∗f1 −h22(δ∗u2 − δ∗u1) = (r1)2, (12.62)

δ∗u2 − δ∗u1 −h22(δ∗v2 − δ∗v1) = (r2)2, (12.63)

δ∗q2 − δ∗q1 −h22(δ∗t2 − δ∗t1) = (r3)2, (12.64)

δ∗q2 − δ∗q1 −h12(δ∗p2 − δ∗p1) = (r4)2 (12.65)

(a1)2δ∗v2 − (a2)2δ

∗v1 + (a3)2δ∗f2 − (a4)2δ

∗f1 + (a5)2δ∗u2 − (a6)2δ

∗u1 = (r5)2,

(12.66)

(b1)2δ∗t2 − (b2)2δ

∗t1 + (b3)2δ∗f2 − (b4)2δ

∗f1 + (b5)2δ∗u2 − (b6)2δ

∗u1

+ (b7)2δ∗s2 − (b8)2δ

∗s1 + (b9)2δ∗v2 − (b10)2δ

∗v1 = (r6)2, (12.67)

(c1)2δ∗p2 − (c2)2δ

∗p1 + (c3)2δ∗f2 − (c4)2δ

∗f1 + (c5)2δ∗q2 − (c6)2δ

∗q1+ = (r7)2.

(12.68)

188

In matrix form put d = −h1

2, we have

0 0 0 −1 0 0 0

0 0 0 0 d 0 0

0 0 0 0 0 d 0

0 0 0 0 0 0 d

0 0 0 (a4)2 (a2)2 0 0

0 0 0 (b4)2 (b10)2 (b2)2 0

0 0 0 (c4)2 0 0 (c2)2

δ∗v0

δ∗t0

δ∗p0

δ∗f1

δ∗v1

δ∗t1

δ∗p1

+

d 0 0 1 0 0 0

−1 0 0 0 d 0 0

0 −1 0 0 0 d 0

0 0 −1 0 0 0 d

(a6)2 0 0 (a3)2 (a1)2 0 0

(b6)2 (b8)2 0 (b3)2 (b9)2 (b1)2 0

0 0 (c6)2 (c3)2 0 0 (c1)2

δ∗u1

δ∗s1

δ∗q1

δ∗f2

δ∗v2

δ∗t2

δ∗p2

+

d 0 0 0 0 0 0

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

(a5)1 0 0 0 0 0 0

(b5)1 (b7)1 0 0 0 0 0

0 0 (c5)1 0 0 0 0

δ∗u2

δ∗s2

δ∗q2

δ∗f3

δ∗v3

δ∗t3

δ∗p3

=

(r1)2

(r2)2

(r3)2

(r4)2

(r5)2

(r6)2

(r7)2

,

189

that is

[B2

] [δ∗1

]+

[A2

] [δ∗2

]+

[C2

] [δ∗3

]=

[r2

]. (12.69)

For J = J − 1

(r1)j−1 = δ∗fj−1 − δ∗fj−2 −hj−1

2(δ∗uj−1 − δ∗uj−2), (12.70)

(r2)j−1 = δ∗uj−1 − δ∗uj−2 −hj−1

2(δ∗vj−1 − δ∗vj−2), (12.71)

(r3)j−1 = δ∗gj−1 − δ∗gj−2 −hj−1

2(δ∗tj−1 − δ∗tj−2), (12.72)

(r4)j−1 = δ∗qj−1 − δ∗qj−2 −hj−1

2(δ∗pj−1 − δ∗pj−2) (12.73)

((a1)j−1δ∗vj−1 − (a2)j−1δ

∗vj−2 + (a3)j−1δ∗fj−1 − (a4)j−1δ

∗fj−2 + (a5)j−1δ∗uj−1

− (a6)j−1δ∗uj−2) = (r5)j−1, (12.74)

((b1)j−1δ∗tj−1 − (b2)j−1δ

∗tj−2 + (b3)j−1δ∗fj−1 − (b4)j−1δ

∗fj−2 + (b5)j−1δ∗uj−1

− (b6)j−1δ∗uj−2 + (b7)j−1δ

∗sj−1 − (b8)j−1δ∗sj−2 + (b9)j−1δ

∗vj−1

− (b10)j−1δ∗vj−2) = (r6)j−1, (12.75)

((c1)j−1δ∗pj−1 − (c2)j−1δ

∗pj−2 + (c3)j−1δ∗fj−1 − (c4)j−1δ

∗fj−2 + (c5)j−1δ∗qj−1

− (c6)j−1δ∗qj−2) = (r7)j−1. (12.76)

190

In matrix form put d = −h1

2, we get

0 0 0 −1 0 0 0

0 0 0 0 d 0 0

0 0 0 0 0 d 0

0 0 0 0 0 0 d

0 0 0 (a4)j−1 (a2)j−1 0 0

0 0 0 (b4)j−1 (b10)j−1 (b2)j−1 0

0 0 0 (c4)j−1 0 0 (c2)j−1

δ∗uj−3

δ∗sj−3

δ∗gj−3

δ∗fj−2

δ∗vj−2

δ∗tj−2

δ∗pj−2

+

d 0 0 1 0 0 0

−1 0 0 0 d 0 0

0 −1 0 0 0 d 0

0 0 −1 0 0 0 d

(a6)j−1 0 0 (a3)j−1 (a1)j−1 0 0

(b6)j−1 (b8)j−1 0 (b3)j−1 (b9)j−1 (b1)j−1 0

0 0 (c6)j−1 (c3)j−1 0 0 (c1)j−1

δ∗uj−2

δ∗sj−2

δ∗qj−2

δ∗fj−1

δ∗vj−1

δ∗tj−1

δ∗pj−1

+

d 0 0 0 0 0 0

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

(a5)j−1 0 0 0 0 0 0

(b5)j−1 (b7)j−1 0 0 0 0 0

0 0 (c5)j−1 0 0 0 0

δ∗uj−1

δ∗sj−1

δ∗qj−1

δ∗fj−2

δ∗vj−2

δ∗tj−2

δ∗pj−2

=

(r1)j−1

(r2)j−1

(r3)j−1

(r4)j−1

(r5)j−1

(r6)j−1

(r7)j−1

,

191

that is

[Bj−2

] [δ∗j−2

]+

[Aj−1

] [δ∗j−1

]+

[Cj−1

] [δ∗j

]=

[rj−1

]. (12.77)

For J = J

(r1)J = δ∗fJ − δ∗fJ−1 −hJ2(δ∗uJ − δ∗uJ−1), (12.78)

(r2)J = δ∗uJ − δ∗uJ−1 −hJ2(δ∗vJ − δ∗vJ−1), (12.79)

(r3)J = δ∗gJ − δ∗gJ−1 −hJ2(δ∗tJ − δ∗tJ−1), (12.80)

(r4)J = δ∗qJ − δ∗qJ−1 −hJ2(δ∗pJ − δ∗pJ−1) (12.81)

((a1)Jδ∗vJ − (a2)Jδ

∗vJ−1 + (a3)Jδ∗fJ − (a4)Jδ

∗fJ−1 + (a5)Jδ∗uJ

− (a6)Jδ∗uJ−1) = (r5)J , (12.82)

((b1)Jδ∗tJ − (b2)Jδ

∗tJ−1 + (b3)Jδ∗fJ − (b4)Jδ

∗fJ−1 + (b5)Jδ∗uJ

− (b6)Jδ∗uJ−1 + (b7)Jδ

∗sJ − (b8)Jδ∗sJ−1 + (b9)Jδ

∗vJ − (b10)Jδ∗vJ−1) = (r6)J ,

(12.83)

((c1)Jδ∗pJ − (c2)Jδ

∗pJ−1 + (c3)Jδ∗fJ − (c4)Jδ

∗fJ−1 + (c5)Jδ∗qJ

− (c6)Jδ∗qJ−1) = (r7)J . (12.84)

192

Matrix form by d = −h1

2is

0 0 0 −1 0 0 0

0 0 0 0 d 0 0

0 0 0 0 0 d 0

0 0 0 0 0 0 d

0 0 0 (a4)J (a2)J 0 0

0 0 0 (b4)J (b10)J (b2)J 0

0 0 0 (c4)J 0 0 (c2)J

δ∗uj−2

δ∗sj−2

δ∗qj−2

δ∗fj−1

δ∗vj−1

δ∗tj−1

δ∗pj−1

+

d 0 0 1 0 0 0

−1 0 0 0 d 0 0

0 −1 0 0 0 d 0

0 0 −1 0 0 0 d

(a6)J 0 0 (a3)J (a1)J 0 0

(b6)J (b8)J 0 (b3)J 0 (b1)J 0

0 0 (c6)J (c3)J 0 0 (c1)J

δ∗uj−1

δ∗sj−1

δ∗qj−1

δ∗fJ

δ∗vJ

δ∗tJ

δ∗pJ

=

(r1)J

(r2)J

(r3)J

(r4)J

(r5)J

(r6)J

(r7)J

,

that is

[BJ

] [δ∗J−1

]+

[AJ

] [δ∗J

]=

[rJ

]. (12.85)

The linearized differential equation of the system (Eq. 12.30 − 12.36) has a block

tridiagonal structure which is

193

[A1

] [C1

][B2

] [A2

] [C2

]·

·

·

·

·

· [Bj−1

] [Aj−1

] [Cj−1

][BJ

] [AJ

]

[δ∗1

][δ∗2

]·

·

·[δ∗j−1

][δ∗J

]

=

[r1

][r2

]·

·

·[rj−1

][rJ

]

,

(12.86)

that is [A

] [δ∗

]=

[r

](12.87)

The entries defined in Eq. (12.88) are

[A1

]=

0 0 0 1 0 0 0

d 0 0 0 d 0 0

0 d 0 0 0 d 0

0 0 d 0 0 0 d

(a2)1 0 0 (a3)1 (a1)1 0 0

(b10)1 (b2)1 0 (b3)1 (b9)1 (b1)1 0

0 0 (c2)1 (c3)1 0 0 (c1)1

, d = −h12

194

[Aj

]=

d 0 0 1 0 0 0

−1 0 0 0 d 0 0

0 −1 0 0 0 d 0

0 0 −1 0 0 0 d

(a6)j 0 0 (a3)j (a1)j 0 0

(b6)j (b8)j 0 (b3)j (b9)j (b1)j 0

0 0 (c6)j (c3)j 0 0 (c1)j

, d = −h12, 2 ≤ j ≤ J

[Bj

]=

0 0 0 −1 0 0 0

0 0 0 0 d 0 0

0 0 0 0 0 d 0

0 0 0 0 0 0 d

0 0 0 (a4)j (a2)j 0 0

0 0 0 (b4)j (b10)j (b2)j 0

0 0 0 (c4)j 0 0 (c2)j

, d = −h12, 2 ≤ j ≤ J

195

[Cj

]=

d 0 0 0 0 0 0

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

(a5)j 0 0 0 0 0 0

(b5)j (b7)j 0 0 0 0 0

0 0 (c5)j 0 0 0 0

, d = −h12, 2 ≤ j ≤ J − 1

[δ∗1

]=

δ∗v0

δ∗t0

δ∗p0

δ∗f1

δ∗v1

δ∗t1

δ∗p1

,

[δ∗J

]=

δ∗uj−1

δ∗sj−1

δ∗qj−1

δ∗fj

δ∗vj

δ∗tj

δ∗pj

2 ≤ j ≤ J,

[rJ

]=

(r1)j− 12

(r2)j− 12

(r3)j− 12

(r4)j− 12

(r5)j− 12

(r6)j− 12

(r7)j− 12

LU method is used to solve Eq.(12.89) and [δ∗] is calculated from this block tridiago-

nal matrix. The solution procedure is repeated until

δ∗v(i)0 ≤ ξ, (12.88)

where ξ = 0.001 is a small positive value.

[A][δ∗] = [r], (12.89)

196

The solution process end when

δ∗v(i)0 ≤ ξ (12.90)

12.4 Discussion

A comparative study for stretching cylinder and flat sheet is made. The convective flow

of Williamson fluid is illustrated through graphs. The solid lines (i.e. γ = 0.0) stand for flat

sheet while dash (i.e. γ = 0.1) lines represent the stretching cylinder. Take γ =Ha = 0.1,

Ec = 0.1, δ = 0.1, Pr = 0.7, Bi = 0.1, K = 0.7, Ks = 0.9 and Sc = 1.2. These

variables are fixed except the variable mentioned in Figures. Table 12.1 shows comparison

of present results with previous published data. The results of dimensionless variables i.e.

curvature variable, Hartman number, Biot number, Prandtl number and others involving

parameters are explained for velocity, temperature and concentration distributions. More-

over the skin friction and Nussult number are disclosed through these parameters.

Fig. 12.3 indicates the influence ϵ on velocity outline. Higher values of ϵ tend to

enhance the velocity near the cylinder while it declines for away from the cylinder. Phys-

ically it means that higher values of ϵ correspond to enhance the relaxation time of liquid

which produces resistance for the liquid flow. Further the velocity via stretching cylinder

is dominated when compared to flat sheet. Fig. 12.4 indicates Hartman number Ha effect

on velocity profile. It is noticed that velocity profile significantly declines for larger Ha.

Larger values of curvature parameter γ enhance the fluid flow (see Fig. 12.5). The velocity

profile first decreases and then increases as the value of γ is enhanced. This is due to the

inverse relation of γ and radius R of cylinder. Infact for larger γ, R of cylinder becomes

less.

197

Fig. 12.6 presents the plots for temperature profile via Hartman number Ha. The tem-

perature profile enhances for larger Ha. It is obvious that larger Ha correspond to decline

in velocity of fluid and there is rise in thermal boundary layer. Heat transfer rate enhances

as the values ofHa is increased. The temperature profile in case of cylinder is more promi-

nent when compared with flat sheet. Fig. 12.7 prepares the plots for temperature via Pr.

Thermal layer thickness decreases for higher Prandtl number. Larger Pr reduce the ther-

mal diffusivity of fluid. Moreover the temperature incase of cylinder is noted higher than

flat sheet. Therefore the temperature decreases. Fig. 12.8 presents consequences of Biot

number Bi on temperature gradient. Higher convection correspond to enhance the surface

temperature. Higher temperature behavior is observed for cylinder than flat sheet. Effect

of heat generation parameter δ on temperature is sketched in Fig. 12.9. The temperature

enhances as the values of δ are increased. Larger values of δ results more heat produced

and consequently the temperature of liquid rises. The heat produces in case of cylinder is

higher when compared with flat sheet.

The plots of homogeneous K and heterogeneous Ks variables for concentration are

displayed in Figs. (12.10,12.11). Here concentration reduces via larger values of homo-

geneous parameter K as well as heterogeneous parameter Ks. The concentration for flat

sheet is noted higher than cylinder. It is noted that solutal layer thickness of reactants en-

hances for higher η and no effect is noted after certain value of η. Plots for concentration

via Sc is focused in Fig. 12.12. Due to inverse relation of Sc with mass diffusivity the con-

centration increases for larger ScThe concentration in case of flat sheet is more prominent

than cylinder.

The skin friction coefficient and magnitude of heat transfer are depicted in Figs. (12.13,

12.13). Figs. (12.13,12.14) are plotted for skin friction via Weissenberg number, Hartman

198

number and curvature parameter. Magnitude of skin friction enhances via larger Hartman

number Ha and curvature parameter γ while it reduces for larger Weissenberg number

ϵ. Skin friction for cylinder is high when compared with flat sheet. Magnitude of heat

transfer increases via larger Pr, γ and Bi (see Figs. 12.15,12.16). The heat transfer rate

enhances via Pr, γ and Bi. Further heat transfer rate in cylinder is noted higher than flat

sheet. Fig. 12.17 represents that higher values of δ and Ec trend to decrease the heat

transfer coefficient. The heat transfer coefficient in case of stretching cylinder is higher

when compared with flat sheet.

199

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f ′ (η

)

2.8 3 3.2 3.4

0.05

0.1

0.15

0.2

0.25

0.3

ε = 0.1, 0.5, 0.9

γ = 0.0

γ = 0.1

Fig. 12.3: Plots via ϵ for f ′(η).

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f ′ (η

)

1 1.2 1.4 1.6

0.15

0.2

0.25

0.3

0.35

0.4

γ = 0.0

γ = 0.1

Ha = 0.1, 0.5, 0.9

Fig. 12.4: Plots via Ha for f ′(η).

200

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f ′ (η

)

γ = 0.1, 0.5, 0.9

Fig. 12.5: Plots via γ for f ′(η).

0 1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

η

θ (

η)

γ = 0.0

γ = 0.1

Ha = 0.1. 0.5, 0.9

Fig. 12.6: Plots via Ha for θ(η).

201

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

θ (

η)

Pr = 0.7, 1.2, 1.5

γ = 0.0

γ = 0.1

Fig. 12.7: Plots via Pr for θ(η).

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

θ (

η)

Bi = 0.1, 0.5, 0.9

γ = 0.0

γ = 0.1

Fig. 12.8: Plots via Bi for θ(η).

202

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

η

θ (η

)

δ = − 1.0, − 0.5, − 0.25, 0.0, 0.1

γ = 0.0

γ = 0.1

Fig. 12.9: Plots via δ for θ(η).

0 1 2 3 4 5 6 7 80.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

η

ζ (

η)

K = 0.4, 0.7, 1

γ = 0.0

γ = 0.1

Fig. 12.10: Plots via K for ζ(η).

203

0 1 2 3 4 5 6 7 80.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ζ (

η) Ks = 0.8, 1.2, 1.5

γ = 0.0

γ = 0.1

Fig. 12.11: Plots via Ks for ζ(η).

0 1 2 3 4 5 6 7 80.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

η

ζ (

η)

γ = 0.0

γ = 0.1

Sc = 0.7, 1.2, 1.5

Fig. 12.12: Plots via Sc for ζ(η).

204

ε

Re x0.

5C

fx

0.1 0.125 0.15 0.175 0.20.925

0.95

0.975

1

1.025

1.05

1.075

1.1

1.125

1.15

Ha= 0.1Ha= 0.2Ha = 0.3Ha = 0.1Ha = 0.2Ha = 0.3

γ = 0.0

γ = 0.1

Fig. 12.13: Plots for skin friction coefficient via ϵ and Ha.

Ha

Re x0.

5C

fx

0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

γ = 0.1

γ = 0.5

γ = 0.9

Fig. 12.14: Plots for skin friction coefficient via Ha and γ.

205

Pr

Re x-0

.5N

ux

0.7 0.725 0.75 0.775 0.8

0.4

0.5

0.6

0.7

0.8

γ = 0.1

γ = 0.5

γ = 0.9

Fig. 12.15: Plots for Nusselt number via Pr and γ.

Βi

Re x-0

.5N

u x

0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

Pr = 0.7Pr = 1.2Pr = 1.5Pr = 0.7Pr = 1.2Pr = 1.5

γ = 0.0

γ = 0.1

Fig. 12.16: Plots for Nusselt number via Pr and Bi.

206

Ec

Re x-0

.5N

u x

0.2 0.4 0.6 0.8 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

δ = -1.0δ = -0.5δ = -0.25δ = -1.0δ = -0.5δ = -0.25

γ = 0.0γ = 0.1

Fig. 12.17: Plots for Nusselt number via Ec and δ.

Table 12.1: Comparison of −f ′′(0) in limiting case via γ when ϵ = Ha = 0.

γ [111] PR0.0 1.0000 1.00000.25 1.094378 1.0943780.5 1.188715 1.1887150.75 1.281833 1.2818331.0 1.459308 1.459308

12.5 Main findings

Computational study for convective flow of Willimson fluid is investigated. The main

outcomes are

• Velocity outline declines for Weissenberg number and Hartman number while it en-

hances for curvature parameter. It is also noted that velocity distribution in case of

cylinder is higher when compared with flat sheet.

• Higher temperature is noted for Biot number while reverse trend is observed for

207

Prandtl number Pr. Temperature in case of cylinder is noted higher than flat sheet.

• Reaction parameters and Schmidt number show opposite behavior for concentra-

tion. Further the concentration in case of flat sheet dominates when compared with

cylinder.

• Skin friction enhances via curvature parameter, Hartman number and it decreases

via Weissenber number. Skin friction for cylinder is higher when compared with flat

sheet.

• Larger Bi, Pr and γ correspond to advance the heat transfer and opposite behavior

is observed for Eckert number, heat generation/absorption parameter. Moreover the

heat transfer rate via cylinder is more than flat sheet.

208

Bibliography

[1] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (1991) 56-58.

[2] P.M Ajayan and S. Iijima, Capillarity-induced filling of carbon nanotubes, Nature

361 (1993) 333-334.

[3] H.C. Choi, S. Kundaria, D. Wang, A. Javey, Q. Wang, M. Rolandi and H. Dai, Effi-

cient formation of iron nanoparticle catalysts on silicon oxide by hydroxylamine for

carbon nanotube synthesis and electronics, Nano Lett. 3 (2003) 157-161.

[4] S. Helveg, C.L. Cartes, J. Sehested, P.L. Hansen, S. Bjerne, R.J. Clausen, Rostrup-

Nielsen, F.A. Pedersen and J.K. Nrskov, Atomic-scale imaging of carbon nanofibre

growth, Nature 427 (2004) 426-429.

[5] S. Hofmann, R. Sharma, C. Ducati, G. Du, C. Mattevi, C. Cepek, M. Cantoro, Can-

toro, S. Pisana, A. Parvez, F. C. Sodi, A.C. Ferrari, R.D. Borkowski, S. Lizzit, L.

Petaccia, A. Goldoni and J. Robertson, In situ observations of catalyst dynamics dur-

ing surface-bound carbon nanotube nucleation, Nano Lett. 7 (2007) 602-608.

[6] D.V. Michael, H.S Tawfick, S.J. Park, D. Copic, Z. Zhao, W. Lu and A.J. Hart, Di-

verse 3D microarchitectures made by capillary forming of carbon nanotubes, Adv.

Mater. 22 (2010) 4384-4389.

209

[7] T. Hayat, Z. Hussain, A. Alsaedi and S. Asghar, carbon nanotubes effects in the

stagnation point flow towards a nonlinear stretching sheet with variable thickness,

Adv. Powder Technol. 27 (2016) 16771688.

[8] R. Haq, F. Shahzad and Q.M. Mdallal, MHD pulsatile flow of engine oil based carbon

nanotubes between two concentric cylinders, Results Phys. 7 (2017) 57-68.

[9] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME-

Publications-Fed 231 (1995) 99-106.

[10] M.R. Hajmohammadi, H. Maleki, G. Lorenzini and S.S. Nourazar, Effects of Cu and

Ag nano-particles on flow and heat transfer from permeable surfaces, Adv. Powder

Technol. 26 (2015) 193-199.

[11] M. Sheikholeslami, M.M. Rashidi and D.D. Ganji, Effect of non-uniform magnetic

field on forced convection heat transfer ofwater nanofluid, Comput. Meth. Appl.

Mech. Eng. 294 (2015) 299-312.

[12] A.S. Dogonchi, K. Divsalar and D.D. Ganji, Flow and heat transfer of MHD

nanofluid between parallel plates in the presence of thermal radiation, Comput. Meth.

Appl. Mech. Eng. 310 (2016) 58-76.

[13] T. Hayat, M.I. Khan, M. Waqas, A. Alsaedi and M. Farooq, Numerical simulation for

melting heat transfer and radiation effects in stagnation point flow of carbon-water

nanofluid, Comput. Meth. Appl. Mech. Eng. 315 (2017) 1011-1024.

[14] M. Sheikholeslami, M.B. Gerdroodbary and D.D. Ganji, Numerical investigation of

forced convective heat transfer of Fe3O4-water nanofluid in presence of external

magnetic source, Comput. Meth. Appl. Mech. Eng. 315 (2017) 831-845.

210

[15] M. Sheikholeslami, Q.M. Zia and R. Ellahi, Influence of induced magnetic field

on free convection of nanofluid considering Koo-Kleinstreuer-Li (KKL) correlation,

Appl. Sci. 6 (2016) 324.

[16] T. Hayat, Z. Hussain and A. Hobiny, Computational analysis for velocity slip and

diffusion species with carbon nanotubes, Results Phys. 7 (2017) 3049-3058.

[17] A. Zeeshan, A. Majeed and R. Ellahi, Effect of magnetic dipole on viscous ferro-fluid

past a stretching surface with thermal radiation, J. Mol. Liq. 215 (2016) 549-554.

[18] A. Majeed, A. Zeeshan and R. Ellahi, Unsteady ferromagnetic liquid flow and heat

transfer analysis over a stretching sheet with the effect of dipole and prescribed heat

flux, J. Mol. Liq. 223 (2016) 528-533.

[19] A. Majeed, A. Zeeshan, S.Z. Alamri and R. Ellahi, Heat transfer analysis in ferro-

magnetic viscoelastic fluid flow over a stretching sheet with suction, Neural Comput.

Appl. DOI: 10.1007/s00521-016-2830-6.

[20] M. Sheikholeslami and R. Ellahi, Three dimensional mesoscopic simulation of mag-

netic field effect on natural convection of nanofluid, Int. J. Heat Mass Transf. 89

(2015) 799-808.

[21] B. Mahanthesh, B.J. Gireesha and R.S.R. Gorla, Nonlinear radiative heat transfer in

MHD three-dimensional flow of water based nanofluid over a non-linearly stretching

sheet with convective boundary condition, J. Nigerian Math. Society 35 (2016) 178-

198.

[22] B. Mahanthesh, B.J. Gireesha, R.S.R. Gorla, F.M. Abbasi and S.A. Shehzad, Nu-

merical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional

211

non-linear stretching surface with prescribed surface heat flux boundary, J. Magn.

Magn. Mater. 417 (2016) 189-196.

[23] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, On magnetohydrodynamic three-

dimensional flow of nanofluid over a convectively heated nonlinear stretching sur-

face, Int. J. Heat Mass Transf. 100 (2016) 566-572.

[24] J.A. Khan, M. Mustafa and A. Mushtaq, On three-dimensional flow of nanofluids

past a convectively heated deformable surface: A numerical study, Int. J. Heat Mass

Transf. 94 (2016) 49-55.

[25] T. Aziz, F.M. Mahomed, A. Shahzad and R. Ali, Travelling wave solutions for the

unsteady flow of a third grade fluid induced due to impulsive motion of flat porous

plate embedded in a porous medium, J. Mech. 30 (2014) 527-535.

[26] C. Sulochana and N. Sandeep, Dual solutions for radiative MHD forced convective

flow of a nanofluid over a slendering stretching sheet in porous medium, J. Naval

Architecture Marine Eng. 12 (2015) 115-124.

[27] J. Ahmed, A. Begum, A. Shahzad and R. Ali, MHD axisymmetric flow of power-law

fluid over an unsteady stretching sheet with convective boundary conditions, Results

Phy. 6 (2016) 973-981.

[28] T. Mahmood, J. Ahmed, A. Shahzad, R. Ali and Z. Iqbal, Convective heat transfer

of viscous fluid over a stretching sheet embedded in a thermally stratified medium,

Bulg. Chem. Commun. 48 (2016) 506-513.

[29] M.J. Babu and N. Sandeep, MHD non-Newtonian fluid flow over a slendering stretch-

ing sheet in the presence of cross-diffusion effects, Alex. Eng. J. 55 (2016) 21932201.

212

[30] T. Hayat, M. Waqas, S.A. Shehzad and A. Alsaedi, Chemically reactive flow of third

grade fluid by an exponentially convected stretching sheet, J. Mol. Liq. 223 (2016)

853-860.

[31] M.J. Babu and N. Sandeep, Effect of variable heat source/sink on chemically reacting

3D slip flow caused by a slendering stretching sheet, Int. J. Eng. Research Africa, 25

(2016) 58-69.

[32] J.V.R. Reddy, V. Sugunamma and N. Sandeep, Effect of frictional heating on radia-

tive ferrofluid flow over a slendering stretching sheet with aligned magnetic field,

European Physical J. Plus 132 (2017) 7.

[33] A. Tamayol, K. Hooman and M. Bahrami, Thermal analysis of flow in a porous

medium over a permeable stretching wall, Trans. Porous Med. 8 (2010) 661-676.

[34] J.T. Hong, Y. Yamada and C.L. Tien, Effect of non-Darcian and nonuniform porosity

on vertical plate natural convection in porous medium, Int. J. Heat Mass Transfer 109

(1987) 356-362.

[35] F. Khani, A. Farmany, M. Ahmadzadeh, A. Aziz and F. Samadi, Analytic solution

for heat transfer of a third grade viscoelastic fluid in non-Darcy porous media with

thermophysical effects, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3867-

3878.

[36] D. Pal and S. Chatterjee, Heat and mass transfer in MHD non-Darcian flow of a mi-

cropolar fluid over a stretching sheet embedded in a porous media with non-uniform

heat source and thermal radiation, Commun. Nonlinear Sci. Numer. Simul. 15 (2010)

1843-1857.

213

[37] T. Hayat, Z. Hussain, B. Ahmed and A. Alsaedi, Base fluids with CNTs as nanoparti-

cles through non-Darcy porous medium in convectively heated flow: A comparative

study, Advan. Powder Tech. 28 (2017) 18551865.

[38] M. Sajid, Z. Abbas, T. Javed and N. Ali, Boundary layer flow of an Oldroyd-B fluid

in the region of stagnation point over a stretching sheet, Can. J. Phys. 88 (2010)

635− 640.

[39] S. Abbasbandy, T. Hayat, A. Alsaedi and M.M. Rashidi Numerical and analytical

solutions for Falkner-Skan flow of MHD Oldroyd-B fluid, Int. J. Num. Meth. Heat

Fluid Flow 24 (2014) 390− 401.

[40] T. Hayat, T. Hussain, S.A. Shehzad and A. Alsaedi, Flow of Oldroyd-B fluid with

nanoparticles and thermal radiation. App. Math. Mech. 36 (2015) 69− 80.

[41] T. Hayat, M. Taseer, S.A. Shehzad and A. Alsaedi, Temperature and concentration

stratification effects in mixed convection flow of an Oldroyd-B fluid with thermal

radiation and chemical reaction. Plos One 10 (2015) e0127646.

[42] M. Ramzan, M. Farooq, M.S. Alhothuali, H.M. Malaikah, W. Cui and T. Hayat, Three

dimensional flow of an Oldroyd-B fluid with Newtonian heating, Int. J. Num. Meth.

Heat Fluid Flo w 25(2015) 68-85.

[43] M. Mustafa, R. Ahmad, T. Hayat and A. Alsaedi, Rotating flow of viscoelastic fluid

with nonlinear thermal radiation: a numerical study, Neural Computing Applications

(2016) 1-7.

[44] T. Hayat, S. Qayyum, A. Alsaedi and M. Waqas, Simultaneous influences of mixed

convection and nonlinear thermal radiation in stagnation point flow of Oldroyd-B

214

fluid towards an unsteady convectively heated stretched surface, J. Mol. Liq. 224

(2016) 811-817.

[45] T. Hayat, T. Muhammad, S.A Shehzad and A. Alsaedi, An analytical solution for

magnetohydrodynamic Oldroyd-B nanofluid flow induced by a stretching sheet with

heat generation/absorption, Int. J. Thermal Sci. 111 (2017) 274-288.

[46] K. Hiemenz, Die Grenzschicht an einem in den gleichfor murgen Flussigkeitsstrom

eingentocuchten graden kreiszylinder, Dinglers polytec. J. 326 (1911) 321− 324.

[47] M. Turkyilmazoglu and I. Pop, Exact analytical solutions for the flow and heat trans-

fer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid. Int. J.

Heat Mass Transfer 57 (2013) 82− 88.

[48] R. Ellahi, A. Riaz, S. Abbasbandy, T. Hayat and K. Vafai, A study on the mixed

convection boundary layer flow and heat transfer over a vertical slender cylinder,

Thermal Sci. 18 (2014) 1247-1258.

[49] M. Nawaz, A. Zeeshan, R. Ellahi, S. Abbasbandy and S. Rashidi, Joules and New-

tonian heating effects on stagnation point flow over a stretching surface by means of

genetic algorithm and Nelder-Mead method, Int. J. Numerical Methods Heat Fluid

Flow 25 (2015) 665-684.

[50] R. Ellahi, M. Hassan, A. Zeeshan and A.A. Khan, The shape effects of nanoparticles

suspended in HFE-7100 over wedge with entropy generation and mixed convection,

App. Nanosci. 6 (2016) 641-651.

[51] R. Ellahi, M. Hassan and A. Zeeshan, Aggregation effects on water base

215

Al2O3nanofluid over permeable wedge in mixed convection, Asia Pacific J. Chem.

Eng. 11 (2016) 179-186.

[52] M. Mamourian, K.M. Shirvan, R. Ellahi and A.B. Rahimi, Optimization of mixed

convection heat transfer with entropy generation in a wavy surface square lid-driven

cavity by means of Taguchi approach, Int. J. Heat Mass Transfer 102 (2016) 544-554.

[53] R.V. Williamson, The flow of pseudoplastic materials, Industrial Eng. Chem. Re-

search 21 (1929) 1108-1111.

[54] S.D. Cramer and J.M. Marchello, Numerical evaluation of models describing non-

Newtonian behavior, American Inst. Chem. Eng. J. 14 (1968) 980-983.

[55] D.V. Lyubimov and A.V. Perminov, Motion of a thin oblique layer of a pseudoplastic

fluid, J. Eng. Phys. Thermophys. 75 (2002) 920-924.

[56] T. Hayat, G. Bashir, M. Waqas and A. Alsaedi, MHD 2D flow of Williamson

nanofluid over a nonlinear variable thicked surface with melting heat transfer, J. Mol.

Liq. 223 (2016) 836-844.

[57] T. Hayat, M. Farooq, A. Alsaedi, and Z. Iqbal, Melting heat transfer in the stagnation

point flow of Powell-Eyring fluid, J Thermophys Heat Transfer 27 (2013) 761− 766.

[58] R. Ellahi and M. Hameed, Numerical analysis of steady flows with heat transfer,

MHD and nonlinear slip effects, Int. J. Num. Methods Heat Fluid Flow 22 (2012)

24− 38.

[59] T. Javed, N. Ali, Z. Abbas and M. Sajid, Flow of an Eyring-Powell non-Newtonian

fluid over a stretching sheet. Chem Eng Commun 200 (2013) 327− 336.

216

[60] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flow of

non-Newtonian nanofluid in a pipe: analytical solutions, Appl Math Modeling 37

(2013) 1451− 1457.

[61] T. Hayat, M. Awais and S. Asghar, Radiative effects in a three-dimensional flow of

MHD Eyring-Powell fluid. J Egyp Math Soci 21 (2013) 379− 384.

[62] T. Hayat, M. S. Anwar, M. Farooq and A. Alsaedi, MHD stagnation Point flow of

second grade fluid over a stretching cylinder with heat and mass transfer. Int. J. of

Nonlinear Sci. and Numer. Simul. 15 (2014) 365-376.

[63] P.A. Razafimandimby and M. Sango, Strong solution for a stochastic model of two-

dimensional second grade fluids. Existence, uniqueness and asymptotic behavior.

Nonlinear Anal. 75 (2012) 4251–4270.

[64] I.G. Baoku, B.I. Olajuwon and A.O. Mustapha, Heat and mass transfer on a MHD

third grade fluid with partial slip flow past an infinite vertical insulated porous plate

in a porous medium. Int. J. Heat Fluid Flow 40 (2013) 81–88.

[65] T. Aziz and F.M. Mahomed, Reductions and solutions for the unsteady flow of a

fourth grade fluid on a porous plate, Appl. Math. Comput. 219 (2013) 9187–9195.

[66] A. Aziz and T. Aziz, MHD flow of a third grade fluid in a porous half space with

plate suction or injection: An analytical approach, Appl. Math. Comput. 218 (2012)

10443–10453.

[67] M. Keimanesh, M. M. Rashidi, A. J. Chamkha and R. Jafari, Study of a third grade

non-Newtonian fluid flow between two parallel plates using the multi-step differential

transform method, Computers Math. Appl. 62 (2011) 2871–2891.

217

[68] M. Pakdemirli, T. Hayat, M. Yurusoy, S. Abbasbandy and S. Asghar, Perturbation

analysis of a modified second grade fluid over a porous plate. Nonlinear Analy, Real

World Appl. 12 (2011) 1774–1785.

[69] R. G. Rahman, M.M. Khader and A.M. Megahed, Melting phenomenon in magneto

hydro-dynamics steady flow and heat transfer over a moving surface in the presence

of thermal radiation. Chinese Physics B 22 (2013) 030202.

[70] L. Roberts, On the melting of a semi-infinite body of ice placed in a hot stream of air,

J. Fluid Mech. 4 (1958) 505–528.

[71] K. Das, Radiation and melting effects on MHD boundary layer flow over a moving

surface, Ain Shams Eng. J. 5 (2014) 1207–1214.

[72] T. Hayat, M. Hussain, M. Awais and S. Obaidat, Melting heat transfer in a boundary

layer flow of a second grade fluid under Soret and Dufour effects, Int. J. Num. Meth.

Heat Fluid Flow 23 (2013) 1155-1168.

[73] T. Hayat, M. Farooq, A. Alsaedi and Z. Iqbal, Melting heat transfer in the stagnation

point flow of Powell-Eyring fluid, J. Thermophy. Heat Trans. 27 (2013) 761-766.

[74] M. Waqas, M. Farooq, M.I. Khan, A. Alsaedi, T. Hayat and T. Yasmeen, Magneto-

hydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear

stretched sheet with convective condition, Int. J. Heat Mass Transfer 102 (2016) 766-

772.

[75] T. Hayat, S. Farooq, A. Alsaedi and B. Ahmad, Hall and radial magnetic field effects

on radiative peristaltic flow of CarreauYasuda fluid in a channel with convective heat

and mass transfer, J. Magnetism Magnetic Materials 412 (2016) 207-216.

218

[76] A. Rauf, S.K. Siddiq, F.M. Abbasi, M.A. Meraj, M. Ashraf and S.A. Shehzad, In-

fluence of convective conditions on three dimensional mixed convective hydromag-

netic boundary layer flow of Casson nanofluid, J. Magnetism Magnetic Materials 416

(2016) 200-207.

[77] T. Hayat, T. Muhammad, B. Ahmad and S.A. Shehzad Impact of magnetic field in

three-dimensional flow of Sisko nanofluid with convective condition, J. Magnetism

Magnetic Materials 413 (2016) 1-8.

[78] I.S. Oyelakin, S. Mondal and P. Sibanda, Unsteady Casson nanofluid flow over a

stretching sheet with thermal radiation, convective and slip boundary conditions,

Alex. Eng. J. 55 (2016) 1025-1035.

[79] J.H. Merkin, Natural-convection boundary-layer flow on a vertical surface with New-

tonian heating, Int. J. heat fluid flow 15 (1994) 392− 398.

[80] T. Hayat, S. Ali, M. Awais and M.S. Alhuthali, Newtonian heating in stagnation point

flow of Burgers fluid, Appl. Math. Mech. 36 (2015) 61− 68.

[81] M.Z. Salleh, R. Nazar and I. Pop, Modeling of free convection boundary layer flow on

a solid sphere with Newtonian heating, Acta Applicandae Mathematicae 112 (2010)

263− 274.

[82] M.Z. Salleh, R. Nazar and I. Pop, Numerical solutions of free convection boundary

layer flow on a solid sphere with Newtonian heating in a micropolar fluid, Meccanica

47 (2012) 1261− 1269.

[83] M.A. Chaudhary and J.H. Merkin, A simple isothermal model for homogeneous-

219

heterogeneous reactions in boundary-layer flow, I Equal diffusivities, Fluid Dynamics

Research 16 (1995) 311-333.

[84] M.A. Chaudhary and J.H. Merkin, A simple isothermal model for homogeneous-

heterogeneous reactions in boundary-layer flow. II Different diffusivities for reactant

and autocatalyst, Fluid dynamics research 16 (1995) 335-359.

[85] J.H. Merkin, A model for isothermal homogeneous-heterogeneous reactions in

boundary-layer flow, Mathematical Computer Modelling 24 8 (1996) 125-136.

[86] M.A. Chaudhary and J.H. Merkin, Homogeneous-heterogeneous reactions in

boundary-layer flow: Effects of loss of reactant, Mathematical computer modelling

24 (1996) 21-28.

[87] M.U. Rahman, M. Khan and M. Manzur, Homogeneous-heterogeneous reactions in

modified second grade fluid over a non-linear stretching sheet with Newtonian heat-

ing, Results Physics https://doi.org/10.1016/j.rinp.2017.06.029.

[88] T. Hayat, Z. Hussain, T. Muhammad and A. Alsaedi, Effects of homogeneous and

heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with

variable surface thickness, J. Mol. Liq. 221 (2016) 1121-1127.

[89] T. Hayat, Z. Hussain, M. Farooq and A. Alsaedi, Effects of homogeneous and het-

erogeneous reactions and melting heat in the viscoelastic fluid flow, J. Mol. Liq. 215

(2016) 749-755.

[90] T. Hayat, Z. Hussain A. Alsaedi and B. Ahmad, Heterogeneous-homogeneous reac-

tions and melting heat transfer effects in flow with carbon nanotubes, J. Mol. Liq.

220 (2016) 200-207.

220

[91] T. Hayat, Z. Hussain, A. Alsaedi and S. Asghar, Carbon nanotubes effects in the

stagnation point flow towards a nonlinear stretching sheet with variable thickness,

Advan. Powder Technol. 27 (2016) 1677-1688.

[92] T. Hayat, Z. Hussain, A. Alsaedi and M. Farooq, Magnetohydrodynamic flow by a

stretching cylinder with Newtonian heating and homogeneous-heterogeneous reac-

tions, PloS one 11 (2016) e0156955.

[93] T. Hayat, Z. Hussain, A. Alsaedi and B. Ahmad, Numerical study for slip flow of

carbonwater nanofluids, Comput. Methods Appl. Mech. Eng. 319 (2017) 366-378.

[94] T. Hayat, Z. Hussain and A. Alsaedi, Influence of heterogeneous-homogeneous reac-

tions in thermally stratified stagnation point flow of an Oldroyd-B fluid, Results Phy.

6 (2016) 1161-1167.

[95] Z. Hussain, T. Hayat, A. Alsaedi and B. Ahmad, Three-dimensional convective flow

of CNTs nanofluids with heat generation/absorption effect: A numerical study, Com-

put. Methods Appl. Mech. Eng. 329 (2017) 40-54.

[96] Q. Xue, Model for thermal conductivity of carbon nanotube based composites, Phys-

ica B 368 (2005) 302-307.

[97] P.K. Kameswaran, S. Shaw, P. Sibanda and P.V.S.N. Murthy, Homogeneous-

heterogeneous reactions in a nanofluid flow due to a porous stretching sheet, Int.

J. Heat Mass Transfer 57 (2013) 465-472.

[98] L. Yang and X. Xu, A renovated Hamilton-Crosser model for the effective thermal

conductivity of CNTs nanofluids, Int. Commun. Heat Mass Transfer 81 (2017) 42-50.

221

[99] W.A. Khan, Z.H. Khan and R. Rahi, Fluid flow and heat transfer of carbon nanotubes

along a plate with Navier slip boundary, Appl. Nanosci. 4 (2014) 633-641.

[100] T.R. Mahapatra and A. Gupta, Heat transfer in stagnation-point flow towards a

stretching sheet, Heat Mass Trans., 38 (2002) 517-521.

[101] S. Pop, T. Grosan and I. Pop, Radiation effects on the flow near the stagnation point

of a stretching sheet, Technische Mechanik, 25 (2004) 100-106.

[102] P. Sharma and G. Singh, Effects of variable thermal conductivity and heat

source/sink on MHD flow near a stagnation point on a linearly stretching sheet, J.

Appl. Fluid Mech., 2 (2009) 13-21.

[103] L. Yang and X. Xu, A renovated Hamilton-Crosser model for the effective thermal

conductivity of CNTs nanofluids, Int. Commun. Heat Mass Transfer 81 (2017) 42-50.

[104] T. Hayat, Z. Hussain, A. Alsaedi and M. Mustafa, Nanofluid flow through a porous

space with convective conditions and heterogeneous-homogeneous reactions, J. Tai-

wan Inst. Chem. Eng. 70 (2017) 119-126.

[105] M.J. Babu and N. Sandeep, 3D MHD slip flow of a nanofluid over a slendering

stretching sheet with thermophoresis and Brownian motion effects, J. Mol. Liq. 222

(2016) 1003-1009.

[106] S.J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differen-

tial equations, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 2003-2016.

[107] T. Javed, N. Ali, Z. Abbas, M. Sajid, Flow of an Eyring-Powell non-Newtonian fluid

over a stretching sheet, Chem. Eng. Commun. 200 (2013) 327− 336.

222

[108] M. Fathizadeh, M. Madani, Y. Khan, N. Faraz, A. Yldrm and S. Tutkun, An effec-

tive modification of the homotopy perturbation method for mhd viscous flow over a

stretching sheet, J. King Saud Univ. Sci. 25 (2013) 107− 113.

[109] T. Fang, J. Zhang and S. Yao, Slip MHD viscous flow over a stretching sheet a exact

solution, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3731− 737.

[110] R. Cortell, A note on magnetohydrodynamic flow of a power-law fluid over a

stretching sheet, App. Math. Computation 168 (2005) 557-66.

[111] R.R. Rangi and N. Ahmad, Boundary layer flow past a stretching cylinder and heat

transfer with variable thermal conductivity, Applied Meth. 3 (2012) 205-209.

[112] H.B. Keller and T. Cebeci, Accurate numerical methods for boundary-layer flows.

II: two-dimensional turbulent flows, AIAA J 10 (1972) 1193-1199.

[113] T. Hayat, Z. Hussain, M. Farooq, A. Alsaedi and M. Obaid, Thermally stratified

stagnation point flow of an Oldroyd-B fluid, Int. J. Non. Sci. Num. Sim. 15 (2014) 77

86.

[114] T.R. Mahapatra and A.S. Gupta, Heat transfer in stagnation-point flow of a microp-

olar fluid towards a stretching sheet, Heat Mass Transfer 38 (2002) 517 521.

[115] R. Nazar, N. Amin, D. Filip and I. Pop, Stagnation point flow of a micropolar fluid

towards a stretching sheet, Int. J. Non-Linear Mechanics 39 (2004) 1227 1235.

223